Harmonic state space modelling of
voltage source converters
James Ernest Ormrod
A thesis submitted in partial fulfilment
of the requirements for the degree of
Master of Engineering
in
Electrical and Computer Engineering
at the
University of Canterbury,
Christchurch, New Zealand.
April 2013
ABSTRACT
The thesis describes the development of a model of the three-phase Voltage Source Converter
(VSC) in the Harmonic State Space (HSS) domain, a Linear Time Periodic (LTP) extension
to the Linear Time Invariant (LTI) state space. The HSS model of the VSC directly captures
harmonic coupling effects using harmonic domain modelling concepts, generalised to dynamically varying signals. Constructing the model using a reduced-order three-phase harmonic signal
representation achieves conceptual simplification, reduced computational load, and direct integration with a synchronous frame vector control scheme.
The numerical switching model of the VSC is linearised to gain a small-signal controlled model,
which is validated against time-domain PSCAD/EMTDC simulations. The controlled model is
evaluated as a STATCOM-type system, exercising closed-loop control over the reactive power
flow and dc-side capacitor voltage using a simple linear control scheme. The resulting statespace model is analysed using conventional LTI techniques, giving pole-zero and root-locus
analyses which predict the dynamic behaviour of the converter system. Through the ability to
independently vary the highest simulated harmonic order, the dependence on the closed-loop
response to dynamic harmonic coupling is demonstrated, distinguishing the HSS model from
fundamental-only Dynamic Phasor models by its ability to accurately model these dynamics.
ACKNOWLEDGEMENTS
Firstly, I would like to extend my thanks to my supervisor Dr Alan Wood, who has been a
pleasure to work with and a font of patience and encouragement.
I would also like to thank the great teachers I have had at the University of Canterbury. Along
with Dr Wood and others, Dr Hamish Laird, Dr Bill Heffernan, and the late Associate Professor
Richard Duke all contributed their insight and fostered my enthusiasm for the field.
My family and friends have made this journey through academic pressures and seismic activity
much more bearable, I thank you all.
This work was made possible by the financial support of the EPECentre and the University of
Canterbury, for which I am grateful.
CONTENTS
ABSTRACT
iii
ACKNOWLEDGEMENTS
LIST OF FIGURES
xiii
LIST OF TABLES
xv
GLOSSARY
CHAPTER 1
CHAPTER 2
CHAPTER 3
v
xvii
INTRODUCTION
1
1.1
Relevance of the Voltage-Source Converter
1
1.2
Motivations for Alternative Models
1
1.3
Research Objectives
2
1.4
Thesis Outline
2
HARMONIC DOMAIN MODELLING
5
2.1
Introduction
5
2.2
Harmonic Signal Representation
6
2.3
Frequency Transfer Matrices
6
2.3.1
Basis FTM Operators
6
2.3.2
FTMs for Time Domain Operations
7
HARMONIC STATE SPACE MODELLING
11
3.1
Introduction
11
3.2
Harmonic State Space
12
3.2.1
LTI State Space
12
3.2.2
Exponentially Modulated Periodic Signals
13
3.2.3
Products of EMP Signals
14
3.2.4
Derivative of EMP Signals
15
3.2.5
Harmonic State Space Equations
15
3.2.6
HSS Properties
16
3.3
Passive Network Modelling
17
viii
CONTENTS
CHAPTER 4
THREE PHASE VSC MODELLING
4.1 Introduction
4.1.1 VSC Applications
4.2 VSC Structure and Operation
4.2.1 Structure and Operation
4.2.2 Dead-time Compensation
4.3 VSC Control
4.3.1 PWM Control
4.3.1.1 Carrier-Based PWM
4.3.1.2 Space Vector Modulation
4.3.2 Vector Control
4.4 Harmonic Domain VSC Model
4.4.1 Three-Phase Signal Representation
4.4.2 Numerical Switching Function Model
4.4.3 Derivation of Three-Phase Transfers
4.4.4 Implementation of LTI Model
4.4.5 Time-Domain Signal Reconstruction
4.4.6 Verification of Individual Transfers
4.4.6.1 PSCAD Model
4.4.6.2 Steady-State Transfers
4.4.6.3 Dynamic Transfers
4.5 Uncontrolled System Model
4.5.1 STATCOM Operation and Control
4.5.2 Uncontrolled System Verification
4.6 Conclusion
21
21
22
22
22
22
23
23
23
24
25
26
26
29
30
37
38
39
39
39
43
46
47
49
50
CHAPTER 5
VSC MODEL WITH CONTROL
5.1 Introduction
5.2 Linearisation Strategy
5.2.1 Small-Signal Model Structure
5.2.2 PWM Sampling Error
5.2.3 Switching Function Linearisability
5.3 Open-Loop Controlled Model
5.3.1 Step Responses in the Time Domain
5.3.2 Step Responses in the State Space
5.3.3 Pole-Zero Analysis of the Open-Loop System
5.4 Closed-Loop Controlled Model Construction
5.4.1 Control Variable Measurement
5.4.1.1 DC Voltage
5.4.1.2 Reactive Power
5.4.2 Control System
5.4.3 Complete Closed-Loop Model
5.4.4 Simulation Procedure
5.5 Closed-Loop Simulation Results
5.5.1 Controlled Model Setpoint Tracking
53
53
54
57
58
60
65
65
69
70
74
74
74
75
77
78
80
81
81
ix
CONTENTS
5.6
CHAPTER 6
5.5.2 Response to Grid Perturbations
5.5.3 Model Order and Harmonic Interaction
5.5.4 Control Parameter Root Loci
5.5.5 Closed-Loop Response to Grid Imbalance
Conclusions
CONCLUSIONS AND FUTURE WORK
6.1 Conclusions
6.2 Future Work
REFERENCES
85
88
92
98
101
103
103
104
107
LIST OF FIGURES
3.1
Figurative expansion of an s-domain signal into multiple harmonic reference frames. 14
3.2
Series RL network.
17
3.3
Series RC network.
19
3.4
Parallel RC network.
20
4.1
Numerical switching function generation showing the reference, carrier, and switching signals.
4.2
30
Spectrum of the switching function for the characteristic harmonics, exhibiting
the roll-off characteristic.
31
4.3
Block diagram of the state-space structure of the uncontrolled model.
37
4.4
PSCAD testbench for individual transfers.
40
4.5
Time-domain comparison of ac voltage transfer from a constant dc bus voltage.
41
4.6
Harmonic comparison of ac voltage transfer from a constant dc bus voltage.
41
4.7
Time-domain comparison of dc current transfer from a positive sequence ac current. 42
4.8
Harmonic comparison of dc current, all even terms.
4.9
Time-domain comparison of transfer from a modulated negative sequence ac current to dc current.
42
44
4.10 Harmonic comparison of transfer from a modulated negative sequence ac current
to dc current.
44
4.11 Time-domain comparison of negative-sequence current transfer by modulated inputs in the -5th and 7th harmonic reference frames.
45
4.12 Harmonic comparison of negative-sequence current transfer by modulated inputs
in the -5th and 7th harmonic reference frames.
4.13 Test network circuit diagram.
45
46
4.14 Single-line diagram of simplified network, without shunt filter or dc-side loss for
the analysis of the VSC system at equilibrium.
4.15 Phasor diagram showing VSC system at equilibrium, supplying reactive power.
47
48
xii
LIST OF FIGURES
4.16 Phasor diagram showing VSC system not at equilibrium after adjusting modulation vector, reducing reactive power supply.
48
4.17 Transient evolution of dc voltage during STATCOM start-up.
51
4.18 Transient evolution of phase a voltage during STATCOM start-up.
51
4.19 Transient evolution of dc voltage following introduction of 10% imbalance at 250
ms.
52
4.20 Transient evolution of dc current following introduction of 10% imbalance at 250
ms.
52
5.1
Block diagram of small-signal open-loop control model.
58
5.2
Spectra of linearised switching functions, normalised to unit control inputs, for
~ base = 0.8 − j0.03
mf = 9, M
60
5.3
Absolute linearisation error for changes in Md
62
5.4
Absolute linearisation error for changes in Mq
63
5.5
Absolute linearisation error for changes in Md and Mq
64
5.6
DC bus voltage following the application of a step control input of Md = 0.01.
66
5.7
DC bus voltage following the application of a step control input of Md = 0.05.
66
5.8
DC bus voltage following the application of a step control input of Md = 0.05,
detailing the steady-state offset.
5.9
67
DC bus voltage following the application of a step control input of Md = 0.01,
detailing the initial transient response.
67
5.10 DC bus voltage following the application of a step control input of Mq = 0.01.
68
5.11 DC bus voltage following the application of a step control input of Mq = 0.05.
68
5.12 Fundamental components of harmonic state-space output.
69
5.13 Fundamental components of harmonic state-space output.
71
5.14 Fundamental components of harmonic state-space output.
71
5.15 Pole-zero map of open-loop controlled system, showing entire system.
72
5.16 Pole-zero map of open-loop controlled system, highlighting dominant poles.
73
5.17 Block diagram of feedback network.
78
5.18 Block diagram of small-signal closed-loop control model.
79
5.19 Quadrature converter current following a change in the Iq setpoint.
82
5.20 DC bus voltage following a change in the Iq setpoint.
82
5.21 DC bus voltage following a change in the Iq setpoint, detailing initial transient
response.
5.22 Quadrature converter current following a larger change in the Iq setpoint.
83
83
LIST OF FIGURES
xiii
5.23 DC bus voltage following a larger change in the Iq setpoint.
84
5.24 Applied control inputs following a larger change in the Iq setpoint.
84
5.25 Quadrature converter current following a 1% step reduction in the grid voltage.
86
5.26 DC bus voltage following a 1% step reduction in the grid voltage.
86
5.27 Quadrature converter current following a 5% step reduction in the grid voltage.
87
5.28 DC bus voltage following a 5% step reduction in the grid voltage.
87
5.29 Quadrature converter current following a change in the Iq setpoint, showing divergent response for varied maximum harmonic order.
89
5.30 DC bus voltage following a change in the Iq setpoint, showing divergent response
for varied maximum harmonic order.
89
5.31 DC bus voltage following a change in the Iq setpoint, showing close agreement for
further increases in harmonic order beyond h=19.
5.32 Pole-zero map of closed-loop system for h=1 (red), h=19 (blue).
90
91
5.33 DC bus voltage following a change in the Iq setpoint, plotted for different values
of Kp .
92
5.34 DC bus voltage following a change in the Iq setpoint, plotted for different values
of Ki .
93
5.35 DC bus voltage and quadrature current following a change in the Iq setpoint,
demonstrating fast settling with a properly tuned controlled.
93
5.36 Pole-zero map of closed-loop system for Kp = 0.5 (blue), Kp = 0.6 (green),
Kp = 0.7 (red).
94
5.37 Pole-zero map of closed-loop system for Kp = 0.5 (blue), Kp = 0.6 (green),
Kp = 0.7 (red), detailing locus of dominant poles.
95
5.38 Pole-zero map of closed-loop system for Ki = 20 (blue), Ki = 30 (green), Ki = 40
(red), detailing locus of dominant poles.
96
5.39 Pole-zero map of tuned closed-loop system, detailing dominant poles.
97
5.40 Quadrature converter current following the introduction of a 5% imbalance.
98
5.41 Quadrature converter current following following the introduction of a 5% imbalance, detailing initial transient response.
99
5.42 D-axis control signal following the introduction of a 5% imbalance, detailing initial
transient response.
5.43 DC bus voltage following the introduction of a 5% imbalance.
99
100
5.44 DC bus voltage following the introduction of a 5% imbalance, detailing initial
transient response.
100
LIST OF TABLES
3.1
The signals of an LTI system.
13
4.1
Uncontrolled test system parameters.
49
5.1
Fundamental component absolute linearisation error (per-unit basis).
61
5.2
Closed-loop test system parameters.
85
GLOSSARY
The following glossary contains a list of the commonly used nomenclature and abbreviations
found in this thesis.
NOMENCLATURE
~
M
switching modulation vector
mf
carrier frequency multiple
=
imaginary component
C
set of complex numbers
R
set of real numbers
Z
set of integers
Md
direct-axis component of modulation vector
Mq
quadrature-axis component of modulation vector
x(t)
time-domain signal
xψ (t)
generalised phase time-domain signal
Γ(t)
vector of harmonic exponentials
X
vector of harmonic coefficients
Xψ
generalised phase ac harmonic vector
X̄
harmonic state space matrix
Ψ
phase-dependent shift matrix
D{·}
diagonal matrix operator
ω0
fundamental angular frequency
<
T {·}
real component
Toeplitz matrix operator
ABBREVIATIONS
EHD
Extended Harmonic Domain
EMP
Exponentially Modulated Periodic
FACTS
Flexible AC Transmission System
xviii
GLOSSARY
FFT
Fast Fourier Transform
FTM
Frequency Transfer Matrix
HSS
Harmonic State Space
HVDC
High Voltage Direct Current
LTI
Linear Time Invariant
LTISS
LTI State Space
LTP
Linear Time Periodic
PLL
Phase-Locked Loop
PWM
Pulse-Width Modulation
STATCOM
Static Synchronous Compensator
TCR
Thyristor-Controlled Reactor
UPS
Uninterruptible Power Supply
VSC
Voltage-Source Converter
Chapter 1
INTRODUCTION
This thesis is concerned with the modeling of the three-phase Voltage-Source Converter (VSC)
using a framework developed in the Harmonic State Space (HSS) domain. This chapter summarizes the motivating factors behind this work.
1.1 RELEVANCE OF THE VOLTAGE-SOURCE CONVERTER
The three-phase VSC can be regarded as the fundamental building block of many grid-connected
power electronic systems. Its ability to produce voltage waveforms comprised of arbitrary magnitudes, frequencies, and phases allows it to be used in many different applications including Flexible AC Transmission System (FACTS) devices, on-line Uninterruptible Power Supply
(UPS) installations, high-efficiency switching controlled rectifiers, and VSC-HVDC transmission
systems.
Continuing development in power semiconductor devices have increased device ratings, lowered
cost, and improved efficiency, in turn enabling an expanding scope of applications. The significant and growing importance of the VSC in electric power systems is a major motivation for
this research. Increased penetration of VSC-based systems is occurring at all levels and scales of
the power system, from converters interfacing renewable generation sources, to FACTS devices
supporting the efficient use of transmission resources, to high-efficiency controlled loads. The
combined effect of increasing power levels and system penetration is a wider scope for interaction between converter systems, and thus the demand for models capable of describing these
interactions, as highlighted in [Arrillaga and Watson 2008].
1.2 MOTIVATIONS FOR ALTERNATIVE MODELS
Time-domain simulation techniques for power electronic systems are technologically mature, with
Electromagnetic Transient programs employing Dommel’s technique in combination with inter-
2
CHAPTER 1
INTRODUCTION
polation algorithms and other subroutines to provide a fully general simulation framework for a
wide class of power systems, including those incorporating switching elements [Dommel 1969].
Although able to compute solutions for arbitrary networks, the resulting time-domain output
is of limited use for the purposes of control analysis, or the study of converter interactions,
as the underlying dynamics are not revealed. As will be discussed in Chapter 5, linearised
state-space control models exist for these converters, but they necessarily do not capture higherorder dynamics such as harmonic coupling effects. As discussed in Chapter 2, a wide variety
of harmonic domain models exist for specific applications such as harmonic penetration studies.
The legacy of these models is a large body of harmonic domain modelling and analysis techniques,
some of which can be generalised to the dynamic HSS.
Modeling techniques using the HSS domain hold the promise of unifying harmonic and control
analyses, while providing accurate transient simulated output. Simple controlled HSS models
have proven their ability to produce closed-loop stability results [Möllerstedt and Bernhardsson 2000] for systems where harmonic interaction affects the control response, and to provide
insight into control design in the presence of system resonances [Orillaza 2011].
1.3 RESEARCH OBJECTIVES
The principle objective of this research is to develop a general-purpose harmonic state space
model of the controlled two-level three-phase voltage source converter, which provides a unified
electrical and control model. In developing the modelling framework for this converter model,
the secondary goal is to advance the state of the art of HSS modelling of FACTS devices, by
articulating a coherent interpretation of HSS signals which allows for the unification of electrical
and control models. Another area for investigation is the application of LTI system analysis
techniques to these HSS models, in order to better characterise closed-loop system performance.
1.4 THESIS OUTLINE
Chapter 2 introduces the harmonic domain as the natural domain for modelling non-linear and
modulating elements in power systems, and defines general concepts and procedures for mapping
time-domain relations in linear time periodic systems to harmonic-domain operations using the
construct of the Frequency Transfer Matrix.
Chapter 3 derives the Harmonic State Space as a periodic extension of the LTI State Space
(LTISS). The properties of LTISS systems are identified, and it is demonstrated that the HSS
formulation retains analogous properties. General procedures for the mapping of LTP systems
1.4
THESIS OUTLINE
3
to self-contained HSS models are developed, and applied to passive LTI networks that will be
later used as elements of an HSS power system model.
Chapter 4 details the operation and control of grid-connected Voltage Source Converters, so as
to inform the development of a suitable HSS modelling framework. The HSS model of the VSC
is developed as a set of reduced-order FTMs, described by a numerical switching model. These
transfers are verified against an equivalent PSCAD/EMTDC model in isolation and as part of
a STATCOM-type system.
Chapter 5 extends the VSC model to include linearised transfers from a synchronous vector
control scheme, giving a dynamic small-signal model. Using the reduced-order three-phase
harmonic domain modelling framework of Chapter 4, measurement concepts for dynamic HSS
signals are developed to link the electrical and control elements. Closed-loop performance of
the dynamic model is evaluated, and LTI analysis techniques are applied to generate discrete
root-loci for control parameters.
Chapter 6 closes the thesis with a review of benefits and limitations of the modelling framework developed here, and suggests a domain of applicability. Potential research directions and
applications are identified.
Chapter 2
HARMONIC DOMAIN MODELLING
2.1 INTRODUCTION
This chapter introduces the techniques of harmonic domain modelling, and how they map to
time-domain operations within generalised Linear Time Periodic (LTP) systems. This will form
the basis for the extension of these techniques to dynamically varying harmonic signals in Chapter 3.
Harmonic domain modelling of power systems provides a systematic way of modelling the coupling between harmonic components due to non-linearities and modulating elements, as reviewed
in [Arrillaga et al. 1995]. The harmonic domain, combined with iterative techniques such as
Newton-Raphson methods, gives a steady-state solution to the interaction of these harmonic
sources.
Harmonic domain models have been successfully applied to modelling low-frequency interactions across HVDC links [Larsen et al. 1989] [Hume et al. 2003]. A comprehensive harmonic
domain model for systems containing PWM-switched converters can be found in [Collins 2006],
which integrates the load-flow, harmonic, and control mismatches into an iterative numerical
model, giving steady-state solutions for unbalanced systems containing a variety of PWM-based
converters.
A harmonic domain model forms a complete description of a system in a periodic steady state.
For power electronic systems, this implies that the switching instants of the modelled devices are
periodically fixed, and that the system has relaxed to a cyclic steady state condition [Love 2007].
It will be seen in Chapter 3 that the harmonic domain solution to an LTP system is the devolution
of the fully dynamic harmonic state space solution to the cyclic steady state.
6
CHAPTER 2
HARMONIC DOMAIN MODELLING
2.2 HARMONIC SIGNAL REPRESENTATION
The foundational principle of harmonic domain modelling is the equivalence of a periodic signal
with its Fourier series expansion [Bracewell 2000]. It is represented here as a vector of harmonic
coefficients, Xk ∈ C
x(t) =
X
Xk ejkω0 t
(2.1)
k∈Z
In practical applications, the vector is truncated to a maximum harmonic order h, which is
practicable if the spectral content of the signal decreases with frequency. By defining a finite
vector of harmonic exponentials, Γ(t), as
h
i
Γ(t) = e−jhω0 t . . . e−j2ω0 t e−jω0 t 1 ejω0 t ej2ω0 t . . . ejhω0 t
(2.2)
the signal may be represented more compactly, as
x(t) = Γ(t)X
(2.3)
The vector X is a constant vector of complex harmonic coefficients, comprised of positive and
negative frequency components, which will be conjugate terms for a real signal x(t), as well as
a zero-frequency term which will be restricted to real values for a real signal.
2.3 FREQUENCY TRANSFER MATRICES
The Frequency Transfer Matrix (FTM) is introduced as a general construct for the representation
of various time-domain operations in the harmonic domain. An output harmonic vector is the
result of the product of an FTM and an input harmonic vector, and through this, represents a
time-domain operation mapping a periodic input to a periodic output. This thesis makes use of
the Operational Matrices formulation of Madrigal and Rico [Madrigal and Rico 2004] to describe
operations on LTP signals. Two basis FTMs are defined and discussed here, which will be used
to construct FTMs to enable a complete system description in the harmonic domain.
2.3.1 Basis FTM Operators
Diagonal Matrix
Given a complex vector {an }n∈[−h,h] , the diagonalising operator D pro-
duces a square FTM of rank 2h + 1, with the diagonal elements of an and zeros elsewhere, as in
2.3
7
FREQUENCY TRANSFER MATRICES
2.4
D = D{an }n∈[−h,h]

a−h

..

.



=
a0



(2.4)

..









.
ah
Toeplitz Matrix
Given a complex vector {an }n∈[−h,h] , the Toeplitz operator T produces a
square FTM of rank h + 1, which has constant diagonal terms.
T = T {an }n∈[−h,h]


a0 a−1 . . . a−h

..
..
.. 

.
.
.
 a1


=
..
..
 ..

.
. a−1 
.


..
.
ah
a1 a0
(2.5)
T k,l = ak−l
2.3.2 FTMs for Time Domain Operations
Two classes of operations on periodic signals can be defined, which together form a complete
description of operations within a periodic system. Those in the first class, that of unary
operations, apply a harmonically decoupled transformation to a harmonic vector input. In
the most general case, these operations are an arbitrary phase and magnitude transformation
between each individual harmonic component of the input and output vectors, and can therefore
model any Linear Time Invariant (LTI) operation on harmonic domain signals. Those in the
second class, that of binary operations, produce a harmonic vector output from two harmonic
vector inputs. This is required to model the time-domain operation of the multiplication of two
inputs. Although the two inputs are mathematically interchangeable, one is selected to become
the operator, and the other the operand, in a choice which reflects the structure of the modelled
system.
Harmonically Decoupled FTM
The general form of the harmonically decoupled transfer
8
CHAPTER 2
HARMONIC DOMAIN MODELLING
is described by multiplication with the diagonalised matrix of 2.4. For the input vector U and
output vector W , the operation describing the decoupled transfer is multiplication with the
diagonally-structured FTM V .
W =VU
V , D{vn }n∈[−h,h]
(2.6)
All LTI operations such as scaling, transformation by harmonic admittance, integration and
differentiation are special cases of this FTM.
Integration FTM
For the periodic input vector U with fundamental angular frequency ω0 ,
the FTM ΛR of the transfer to its time integral is described as:
Z
y(t) =
u(t) dt
1
Un
jnω0
1
R
Λ ,D
jnω0 n∈[−h,h]
Yn =
Differentiation FTM
(2.7)
For the periodic input vector U with fundamental angular frequency
ω0 , the FTM ΛD of the transfer to its time derivative is described as:
du(t)
dt
Yn = jnω0 Un
y(t) =
ΛD , D{jnω0 }n∈[−h,h]
(2.8)
Phase Shift FTM
A time-domain shift of the periodic signal U by τ seconds is equivalent
τ
ω0
radians at the fundamental frequency, and therefore nφ radians at every
to a shift by φ =
nth
harmonic component, which may be represented in the following decoupled FTM:
2.3
9
FREQUENCY TRANSFER MATRICES
y(t) = u(t +
φ
)
ω0
Yn = ejnφ Un
ΛΦ (φ) , D{ejnφ }n∈[−h,h]
Three-Phase Shift FTM
(2.9)
A special case of the phase shift FTM can be applied in the
modelling of three-phase power systems. Phase signals (currents or voltages) in balanced and
periodic three-phase systems are identical, but shifted in time by one third of the fundamental
period. This implies that any of the three phase signals may be described as time-shifted
variations of the reference phase a, or equivalently in the harmonic domain as the harmonic
vector of a modified by the harmonic-dependent phase shift defined in equation 2.9.
The positive sequence phase shift ψ relative to phase a is therefore defined as ψa = 0, ψb =
and ψc =
2π
3
−2π
3
over the three phases. The general vectorial representation of this FTM is gained
by applying this phase-dependent shift ψ to the general phase shift FTM:
Ψ = ΛΦ (ψ)
(2.10)
Ψ , D{ejnψ }n∈[−h,h]
(2.11)
In this manner, the FTM may be applied to give a generalised harmonic domain expression for
any phase U ψ in a symmetric system, based on the single harmonic domain vector for the phase
a signal, U a :
U ψ = ΨU a
Multiplication FTM
(2.12)
Implementing multiplication of periodic signals represented as har-
monic vectors is achieved by exploiting the convolution theorem, which states that the Fourier
series expansion of the time-domain product of two signals v(t) and u(t) is equal to the convolution of their respective Fourier series expansions, the harmonic vectors V and U .
w(t) = v(t)u(t)
h
X
k=−h
Wk ejkω0 t =
h
X
m=−h
Vm ejmω0 t
(2.13)
h
X
n=−h
Un ejnω0 t
(2.14)
10
CHAPTER 2
HARMONIC DOMAIN MODELLING
The Toeplitz structure rationalises the product of sums to a matrix multiplication.
Γ(t)W = Γ(t)T {V }m∈[−2h,2h] U
W = T {V }m∈[−2h,2h] U
(2.15)
The term selected as the multiplicand, from which the FTM V is formed, requires harmonic
content of twice the order of the input vector to fully populate the FTM. In the terms of the
positive and negative spectral representation, this is required to model in-band transfers from
an input harmonic component at pω0 to an output harmonic component at −qω0 for p, q ≤ h
(and likewise from −pω0 to qω0 ).
Chapter 3
HARMONIC STATE SPACE MODELLING
3.1 INTRODUCTION
This chapter introduces the HSS as a general framework for the solution of LTP systems, and
shows how it can be applied to power system modelling. It will demonstrate that existing
harmonic domain modelling techniques are a subset of the HSS formulation, and thus the unifying
potential of the HSS framework.
The HSS was developed by Wereley [Wereley 1991] as a framework for the analysis of LTP
systems. The target of Wereley’s analysis was the modelling of helicopter rotor blades, which
exhibit periodic dynamics as they rotate alternately into and out of the direction of travel,
driving a harmonic response which the control system must be designed for. The helicopter
blade system has a clear analogy to a power system containing switching converters, in which a
dominant fundamental frequency input gives rise to harmonic interaction through the switching
action of the converter.
This mode of analysis has its roots in earlier methods for the analysis of periodic systems. Floquet
theory [Floquet 1883], introduced in 1883, provides a time-dependent change of coordinates
for linear systems with periodically time-varying coefficients, resulting in a linear system with
constant coefficients in the new reference frame. Importantly, all the dynamics of the original
system are preserved in the modified reference frame, and stability results can be demonstrated
from within the modified frame. This technique was used by Hill [Hill 1886] for the analysis of
celestial objects, in which periodically varying dynamics arise through their orbits. The common
idea is to use a modified reference frame in which periodic dynamics become constant. In Floquet
theory, this is achieved through a factorisation of the response into the product of a periodic
transition matrix with an exponential response [Alvarez Martin et al. 2011]. In the HSS, this is
achieved through an extension of the conventional LTISS by expanding its domain of signals to
explicitly include harmonic components.
Other authors reach a similar formulation to the HSS under the names of the Dynamic Harmonic
12
CHAPTER 3
HARMONIC STATE SPACE MODELLING
Domain (DHD) [Chavez 2010] or Extended Harmonic Domain (EHD) [Rico et al. 2003]. In this
thesis, the terminology of the HSS is used in acknowledgement of the original formulation, and
also in light of the goal of extending state-space analysis to a harmonic model of the VSC.
3.2 HARMONIC STATE SPACE
The Harmonic State Space is an extension of the conventional LTISS to describe LTP systems.
The LTISS is first characterised here, so that the HSS can be logically extended from it.
3.2.1 LTI State Space
An LTI system may be modelled as a set of first-order linear differential equations of its state
variables, allowing the formulation of a compact algebraic representation as an LTISS system.
ẋ(t) = Ax(t) + Bu(t)
(3.1)
y(t) = Cx(t) + Du(t)
(3.2)
The unilateral Laplace transform provides an integral transformation of the signals x(t) of the
LTISS system, and thus of the state-space formulation itself, with the use of the differential
operator s giving a completely algebraic representation of the LTISS within the s-domain. The
algebraic description allows analysis of the dynamic behaviour of the system through mechanisms
such as the s-domain pole-zero and root-locus techniques.
x(s) = L {x(t)} =
Z
∞
e−st x(t) dt
(3.3)
0−
sx(s) = Ax(s) + Bu(s)
(3.4)
y(s) = Cx(s) + Du(s)
(3.5)
The basis function for the LTISS is the kernel function of the Laplace operator, e−st , where
s = σ + jω. It is by extension of this basis function into harmonic components that the HSS
gains the power to model periodic systems. The integrability of signals on the region of [0, ∞)
determines the set of signals that can be described in the s-domain. This set, the complex
frequency or s-domain, maps to all physically realizable signals of an LTI system, with the
3.2
13
HARMONIC STATE SPACE
exception of the resonant response to an undamped mode [Love 2007]. The form of the signals
of an LTI system in the s-domain are shown in Table 3.1, where u0 ∈ Rp×1 .
Table 3.1
The signals of an LTI system.
LTI Signal
Symbol
Formula
Input
Forced response (steady-state response)
Transient response
Steady state output
Transient output
System transfer function
u(t)
xss (t)
xtr (t)
yss (t)
ytr (t)
H(s)
u0 est
(sI − A)−1 Bu0 est
eAt (x(t0 ) − xss (t0 ))
Cxss (t)
Cxtr (t)
C(sI − A)−1 B
These equations reveal several important qualities of the LTI system. The response of the LTI
system to an s-domain input is seperable into two parts; the transient response, which evolves at
the eigenvalues of the system A matrix, and the forced or steady-state response, which evolves
at the frequencies of the inputs. An important property of such a system is the relaxation of
the transient response. If the system is asymptotically stable, as can be determined by the
eigenvalues of the A matrix, then the transient response tends towards zero as t → ∞. This
property allows for the calculation of the steady state response as a direct algebraic solution of
the state space description of the system for its input, a property that should be retained in any
expanded framework.
In the steady state, the frequencies of the input are those of the output, but transformed through
the gain and phase modifications of the mapping between inputs and outputs, as represented by
the system transfer function. The linear separability of those transfers is confirmed by the equivalent representation of the p-input, q-output system as p.q distinct s-domain transfer functions,
though with the loss of generality and manipulability afforded by the algebraic representation.
This general property of linearity, in which input frequencies map one-to-one with output frequencies, assures that an LTI system model in any single reference frame cannot describe a
modulator, which couples signals at multiple frequencies through time-variant action, but hints
at the expression of a domain which can.
3.2.2 Exponentially Modulated Periodic Signals
By introducing a periodic extension to the kernel function of the Laplace transform, a new signal
representation is gained, the Exponentially Modulated Periodic (EMP) signal:
z(t) =
X
k∈Z
Z k e sk t
(3.6)
14
CHAPTER 3
HARMONIC STATE SPACE MODELLING
where sk = s + jkω0 , giving
z(t) = est
X
Zk ejkω0 t
(3.7)
k∈Z
This signal representation is the product of the periodic signal of 2.1 and the complex exponential
basis of the s-domain. Each component of the signal is modulated exponentially at an integer
multiple of the fundamental frequency. In doing so, this creates a multiplicity of harmonic
reference frames as illustrated in Fig. 3.1, allowing any fixed periodic signal to be represented
by a stationary vector Z in the degenerate case where s = 0, and also dynamically time-varying
signals in the general case.
Unlike the conventional s-domain, the dynamic signal representation is non-unique, because of
the multiplicity of equivalent signals. Any s-domain signal Z0 est can be represented in any k-th
order harmonic reference frame ejkω0 as Zk e(s−jkω0 )t
This signal can be equivalently expressed using the harmonic vector of 2.2, as:
z(t) = est Γ(t)Z
..
.
(3.8)
e(s+j2ω0 )t
e(s+jω0 )t
est
e(s)t
⇐⇒
e(s−jω0 )t
e(s−j2ω0 )t
..
.
Figure 3.1
Figurative expansion of an s-domain signal into multiple harmonic reference frames.
3.2.3 Products of EMP Signals
This signal representation retains the multiplication-convolution duality of the static harmonic
domain representation. This allows for the same implementation of convolution as a matrix
product of a Toeplitz-structured matrix of the harmonic coefficients of the multiplicand,
3.2
15
HARMONIC STATE SPACE
y(t) = A(t)z(t)
st
e Γ(t)Y = est Γ(t)T {A}Z
Y = AT Z
(3.9)
(3.10)
where the shorthand AT is equal to the Toeplitz operator of 2.5 applied to the harmonic vector
A.
3.2.4 Derivative of EMP Signals
In order to form an explicit description of the state-space dynamic equations in terms of harmonic
vectors, it is required to define the derivative of the harmonic modulation vector Γ(t):
Γ̇(t) = Γ(t)N
(3.11)
where N is the diagonally structured derivative matrix of 2.8. Using this form, the derivative
of the EMP signal of 3.8 can be generated using the product rule as:
ż(t) = est Γ(t)sZ + est Γ(t)N Z
(3.12)
3.2.5 Harmonic State Space Equations
The state-space description of a LTP system is written below, in which the system matrices
A(t), B(t), C(t) and D(t) are T0 -periodic functions.
ẋ(t) = A(t)x(t) + B(t)u(t)
(3.13)
y(t) = C(t)x(t) + D(t)u(t)
(3.14)
By expressing the dynamic state-space equations of 3.13 and 3.14 using the EMP signals and
operators defined in 3.8-3.12, the dynamic Harmonic State Space equations are produced as
follows:
16
CHAPTER 3
HARMONIC STATE SPACE MODELLING
est Γ(t)sX + est Γ(t)N X = est Γ(t)AT X + est Γ(t)B T U
sX + N X = AT X + B T U
sX = (AT − N )X + B T U
(3.15)
est Γ(t)Y = est Γ(t)C T X + est Γ(t)D T U
Y = C T X + DT U
(3.16)
These equations map the time-periodic LTP system matrices to their fixed HSS equivalents,
defining the general procedure for the translation of a pure LTP system into an HSS system by
means of the Toeplitz transform and time derivative matrix.
Ā = AT − N
(3.17)
B̄ = B T
(3.18)
C̄ = C T
(3.19)
D̄ = D T
(3.20)
Of note is the presence in the system dynamic matrix of equation 3.15 of the −N X term, which
can be interpreted as forcing a rotation of the nth component of the harmonic state variable at
e−jnω0 , which is equivalent to the reality that in the stationary reference frame, the unforced
state variables will remain constant.
3.2.6 HSS Properties
Just as a stable LTISS system was defined to relax to a steady-state condition for a constant
input vector, in this form it can be seen that the HSS system relaxes to a similar steady-state
condition, specifically the cyclical or harmonic steady state where sX = 0. From this, the
steady-state solution to an arbitrary harmonic input can be solved through manipulation of the
state-space matrices:
3.3
17
PASSIVE NETWORK MODELLING
sX = ĀX + B̄U
Y = C̄X + D̄U
0 = ĀX + B̄U
X = −Ā
−1
B̄U
(3.21)
−1
(3.22)
Y = (−C̄ Ā
B̄ + D̄)U
3.3 PASSIVE NETWORK MODELLING
The HSS modelling techniques described above are applied here to LTI component networks
that will later be used as part of a larger VSC system model. The result is a self-contained HSS
block suitable for the modular system construction described in Section 4.4.4.
Series RL Network
v1
L il (t)
Figure 3.2
R
v2
Series RL network.
As a linear network, describable using first-order differential equations, the series RL network
as shown in Fig. 3.2 can be written in state-space form, using the following state mapping:
h
i
x(t) = il (t)
h
i
y(t) = il (t)
"
#
v1 (t)
u(t) =
v2 (t)
h ˙ i h ih
i h
+ L1
il (t) = −R
i
(t)
l
L
h
i h ih
i h
=
il (t)
1 il (t) + 0 0
(3.23)
(3.24)
(3.25)
"
#
i v (t)
1
−1
L
v2 (t)
"
#
i v (t)
1
v2 (t)
(3.26)
(3.27)
When these time-invariant matrices are generalised as constant LTP matrices, the resultant
18
CHAPTER 3
HARMONIC STATE SPACE MODELLING
Toeplitz transforms of their spectra are constant diagonal matrices of dimension n = [−h, h],
which when combined with the LTP to HSS mapping procedure of equations 3.17-3.20 gives
the HSS frequency transfer matrices and associated harmonic state vectors, resulting in a selfcontained state-space block with defined harmonic inputs and outputs:
Ā = AT − N

−R
L




=





..

.
−R
L




B̄ = 




.
−R
L
−R
L
..
..
0
..
..
.
..
.
..

1
 .

..


C̄ = 
1


..

.










.
−1
L
1
L
.
+jhω0
−R
L
..









.

−R
L
1
L

+ jhω0




Ā = 





..

−jhω0
 
 
 
 
−
 
 
 
 
− jhω0

.
−1
L
.
(3.28)
..
.
−1
L
1
L









(3.29)










1

0
 .

..


D̄ = 
0


..

.

(3.30)

0
..
0
.





0


..
. 

0
(3.31)
3.3
19
PASSIVE NETWORK MODELLING
h
iT
X = . . . Il,−1 Il,+0 Il,+1 . . .
h
iT
Y = . . . Il,−1 Il,+0 Il,+1 . . .
hh
i h
iiT
U = . . . V1,−1 V1,+0 V1,+1 . . .
. . . V2,−1 V2,+0 V2,+1 . . .
(3.32)
(3.33)
(3.34)
The HSS models of these LTI networks are diagonal in structure, the individual harmonic components therefore being independent of each other. This implies these HSS constructions are
not dependent on any specific three-phase signal representation or choice of modelled harmonics.
They can therefore be constructed for any subset of harmonic components, as included in the
state space vectors and matrices.
Series RC Network
vout
R
iin (t)
+
vc (t)
−
Figure 3.3
Series RC network.
The series RC network, as illustrated in Fig. 3.3, has the following state-space form which
similarly translates into a decoupled HSS system:
h ˙ i h ih
i h ih
i
vc (t) = 0 vc (t) + C1 iin (t)
h
i h ih
i h ih
i
vout (t) = 1 vc (t) + R iin (t)
(3.35)
(3.36)
Parallel RC Network
The parallel RC network, as illustrated in Fig. 3.4, has the following state-space form which
similarly translates into a decoupled HSS system:
20
CHAPTER 3
HARMONIC STATE SPACE MODELLING
vout
iin (t)
+
vc (t)
R
−
Figure 3.4
Parallel RC network.
i h ih
i
h ˙ i h ih
−1
vc (t) + C1 iin (t)
vc (t) = RC
h
i h ih
i h ih
i
vout (t) = 1 vc (t) + 0 iin (t)
(3.37)
(3.38)
Chapter 4
THREE PHASE VSC MODELLING
4.1 INTRODUCTION
This chapter introduces the three-phase VSC as the subject of analysis in the HSS domain.
The principles of the operation and control of pulse-width modulated (PWM) VSC systems are
introduced generally and then used to describe a harmonic domain modelling framework suited
for three-phase vector control.
The switching action of the VSC is described using the numerical switching function model
developed in Section 4.4, at first using a fixed control vector and switching function, which
is extended to modelling varying control inputs in Chapter 5. The output of the numerical
switching function model is then used in Section 4.4.3 to define the harmonic-domain transfers
which describe the uncontrolled VSC in the HSS.
Existing harmonic domain models of VSCs usually make use of analytic derivations of the switching spectra, defined as Fourier coefficients of pulse trains as determined by a set of switching
instants [Collins 2006], where control variation is an element of the steady-state solution combining load-flow and harmonic solutions. To date most HSS (or equivalently EHD/DHD) models of
VSCs similarly model fixed switching functions, using individual phase networks to provide the
generality for the modelling of non-characteristic harmonics and unbalanced system conditions
[Madrigal et al. 2000] [Rico et al. 2003].
The goal of the VSC model developed in this chapter is to provide a flexible framework which
is organised around the principles of vector control, and which satisfies a unified interpretation
of three-phase quantities in the harmonic state space, allowing for subsequent extension to a
controlled model.
22
CHAPTER 4
THREE PHASE VSC MODELLING
4.1.1 VSC Applications
The Voltage Source Converter is a highly versatile topology. By modulating a dc bus with
three sets of switches, it acts as a synthetic voltage source of variable magnitudes, phases and
frequencies. As a controllable voltage source, it can indirectly control current and power flows
between the ac and dc terminals. As discussed in Section 4.3, with a suitable control scheme
this allows for the decoupled control of real and reactive power flows, allowing the VSC to be
controlled as a virtual machine.
As an intermediating link between ac and dc systems, this ability to control decoupled power
flows gives the VSC use as a grid-tie inverter for dc energy sources such as photovoltaic power
systems, as a STATCOM, or a controlled rectifier. When two converters are combined with a
shared dc link, the same structure can serve as a variable-speed machine drive, or VSC-HVDC
link.
4.2 VSC STRUCTURE AND OPERATION
4.2.1 Structure and Operation
The fully-controlled two-level three-phase bridge forms the kernel of the three-phase voltagesource converter. It consists of three converter legs, each connecting an ac phase to the positive
and negative rails of the dc bus through a pair of fully-controlled switching devices. These
devices may be of any force-commutated type, including MOSFETs, IGBTs or GTOs, and may
be connected in any combination of parallel or series configurations in order to achieve higher
current or voltage ratings, so long as they are switched unitarily to give the effect of a monolithic
switching device.
The switching devices are controlled between an on-state (conducting) and an off-state (blocking). When the upper switching device is on, the phase is connected to the top rail of the dc
bus, and similarly for the lower switching device. The state of the switches on an individual leg
may be described by a switching function for each phase, sψ (t), which takes the value 1 when
the top switch is conducting, and −1 when the bottom switch is conducting.
4.2.2 Dead-time Compensation
The switching devices are paired with antiparallel diodes such that there always exists a path
for the phase current, even when both switching devices in a leg are off. This state necessarily
occurs during the dead-time period, where both devices on a leg are simultaneously driven off,
to allow the previously conducting device time to commutate to its full off-state blocking voltage
4.3
VSC CONTROL
23
before the alternate device is switched on, in order to prevent simultaneous conduction and the
catastrophic shoot-through that would accompany it. During this dead-time period, the state of
conduction in either the upper or lower antiparallel diode, and thus which dc rail is connected to
the phase, is dependent on the direction of the current in the phase. If left unchecked, this will
result in a departure of the effective switching function from the ordered switching signal, and
thus in the terms of angular or vector control schemes, a departure from the ordered fundamental
frequency phase vector.
Various schemes are available for the active compensation of this effect, the net result of which
is an effective switching function which matches the ordered switching signal. The converter
model considered in this thesis assumes the implementation of one of these schemes, such that
the effect of the dead-time may be ignored in the modelling of both the converter and control
system [Holmes and Lipo 2003]. This assumption is justified based on the capabilities of these
schemes to be transparently integrated into the control system.
4.3 VSC CONTROL
The versatility and controllability of the fully-controlled VSC comes from the ability to apply
arbitrary switching functions to its individual converter legs. This enables a control bandwidth
in proportion to the maximum switching frequency of the devices, which allows for the fundamentally high-bandwidth applications such as active harmonic filtering, as well as a fast response
in fundamental frequency control applications. It is these fundamental frequency control applications which this thesis is concerned with, in which the objective is control over fundamental
frequency voltages, currents, and power flows.
This section introduces the converter control techniques required for basic fundamental frequency
control applications, and their treatment in the harmonic domain model of the VSC.
4.3.1 PWM Control
4.3.1.1
Carrier-Based PWM
Carrier-based PWM is the oldest and most straightforward technique for the translation of a
control reference signal into a switching function, well-described in sources such as [Mohan and
Undeland 2007] and [Holmes and Lipo 2003]. The switching function for an individual phase
sψ (t) is produced from the comparison of two signals, the carrier signal vc (t) and reference signal
vref −ψ (t). The carrier signal is a sawtooth or triangular waveform of frequency ωc , which must
be an integer multiple mf of the fundamental frequency ω0 to produce a switching function with
a fundamental period T0 . In single-phase applications, it is sufficient for mf to be an odd number
24
CHAPTER 4
THREE PHASE VSC MODELLING
so as to avoid the production of even harmonics. For three-phase converters, mf is chosen as an
odd third multiple [Mohan and Undeland 2007].
The reference signal is in the most basic case a set of three sinusoids with the desired amplitude,
or modulation index ma and phase. For the generation of a balanced three-phase switching
~ in either a static or synchronous reference frame.
function, reduces to a single space vector M
The resultant switching signal, sharing the fundamental period, may be expressed as its Fourier
series expansion, and equivalently as a harmonic vector:
sψ (t) =
X
Sψ,k ejkω0 t
(4.1)
k∈Z
sψ (t) = Γ(t)S ψ
(4.2)
When analysed in the frequency domain, the resulting switching signal in the ideal case has a
fundamental frequency component with an identical magnitude and phase to the reference signal.
This is equivalent to stating that the transfer function from ma to |Sψ1 | has a linear gain. There
are two criteria to be satisfied for this linearity to be achieved. Firstly, the reference signal
must be constrained to the domain (−1, 1) over the entire period, or the switching signal will
enter an overmodulation state, resulting in not only a loss of linear gain, but the introduction of
low-order harmonic content. For sinusoidal PWM, this requires that the modulation index ma
be less than 1. Secondly, the modulation frequency ratio must be sufficiently high as to present
a reference waveform evolving linearly compared to the slope of the triangular carrier, in order
to give a linear fundamental component transfer function. It will be seen in Section 4.4 that the
numerical model developed here is equally capable of describing operation in the overmodulation
region.
4.3.1.2
Space Vector Modulation
Space Vector Modulation is a newer technique for the synthesis of a three-phase PWM waveform,
amenable to digital implementation. It maps a scheduled output voltage space vector in the α−β
reference frame to a time-weighted combination of the eight possible switching states (and thus
instantaneous output space vectors) of the three-phase two-level bridge. For a given vector, the
weighted combination of switching states is non-unique because of the choice of utilisation of
the two zero vectors, [000] and [111], which are equivalently zero-valued in the zero-sequence
agnostic α − β reference frame [Kolar et al. 1991].
It has been demonstrated that one of these sub-types of SVM, symmetrical SVM with equal
use of the zero vectors, is equivalent to carrier-based PWM with a 3rd harmonic triangle wave
4.3
25
VSC CONTROL
superimposed on the reference signal [Zhou and Wang 2002]. Whether achieved in the inherently
zero-sequence agnostic SVM framework or augmented to a carrier-based system, the result is
√
an extension to the linear modulation range to ma ≤ 2/ 3 ≈ 1.154, a 15% improvement in
maximum ac voltage, and thus dc bus utilisation, which has now been widely applied. This
equivalence allows SVM schemes to be easily implemented in the numerical switching model
described in Section 4.4, through the modification of the reference signal with a phase-matched
triangular waveform, or sinusoid for THIPWM.
4.3.2 Vector Control
Vector control in VSCs, and power electronic systems in general, allows measured power system
signals to be represented as maximally decoupled quantities, suitable for treatment as independent control variables. This decoupling takes two primary forms, both achieved through the
main tools of vector control, the dq0 and αβγ transforms, Park’s and Clark’s transformations
respectively. The magnitude-preserving form of Park’s transformation is given as:

 
 
2π
cos(θ)
cos(θ − 2π
v
)
cos(θ
+
)
vd
3
3
  a
  2
vq  = − sin(θ) − sin(θ − 2π ) − sin(θ + 2π )  vb 
  3
3
3  
1
1
1
vc
v0
2
2
2
(4.3)
The first form of decoupling achieved through the application of Park’s transformation is the
conversion of the balanced component of the measured three-phase fundamental frequency signal
to a static vector in the synchronous reference frame, where θ = ωsys t+φ. This decouples changes
in the phase of the vector from the periodic rotation of the vector, assuming a constant system
frequency, and reduces the signal dimension from three scalars to a single 2-vector. The second
form of decoupling is achieved when the phase angle φ matches that of the remote voltage source,
which may be the grid voltage at the point of common connection, or an electrical machine EMF
(by definition the direct axis) in drive applications. In this synchronous reference frame, the
axial d and q components of the converter current resolve directly to real and reactive power flow
(or equivalently to torque and flux for a machine drive), and when the remote voltage source
and the controllable voltage source of the converter are connected through a predominantly
inductive impedance, these components may be effectively controlled through the quadrature
manipulation of converter voltage.
This introduces the problem of achieving an accurate and stable reference for the remote voltage,
which may vary in phase and frequency, so as to provide the correct signal φ, as any error in this
signal will translate to an angular error in the reference frame, and all control signals developed
within it. In practise, this is commonly achieved with a phase-locked loop (PLL) mechanism,
which functions as an observer of θ and controls the error between the reference oscillator and
26
CHAPTER 4
THREE PHASE VSC MODELLING
the external measurement. The STATCOM model presented by Wood and Osauskas in [Wood
and Osauskas 2004] embeds a linearised PLL model.
As a harmonic domain model, absolute periodicity at the nominal system frequency must be
assumed, which allows ωsys to be fixed at ω0 . Furthermore, the phase reference is assumed to
be similarly absolute, fixing the synchronous reference frame as an integral part of the model,
resulting in a global fixed reference frame, and effectively ignoring PLL dynamics. This has
obvious implications when modelling power systems subject to a remote source with a varying
phase, which will be dealt with appropriately.
4.4 HARMONIC DOMAIN VSC MODEL
Using the general principles of harmonic domain modelling introduced in Chapter 2, and the
specific considerations of VSC construction and control detailed in this chapter, the harmonic
domain model of the VSC is constructed in this section in accordance with the interpretation
of EMP signals for polyphase applications developed below. The numerical switching function
model is developed to provide the link between the specified operating point and the switching
spectrum that forms the harmonic-domain FTMs.
4.4.1 Three-Phase Signal Representation
Key to using a harmonic domain formulation for the description of a three-phase power system
comprised of LTP elements is the choice of signal representation for three-phase signals. Threephase electric power systems are the target of our analysis, as they dominate the landscape
for the generation, transmission and utilisation of electrical energy at high power levels. The
three-phase system is the simplest polyphase system which allows energisation in a sequence
creating a rotating vector in the phase plane. It is this particular sequence of energisation, as
revealed by the linear separation into the symmetrical components of Fortescue [Fortescue 1918],
that imbues the three-phase power system with the privileged reference frame associated with
the fundamental frequency phase vector, and which should then be considered explicitly when
developing a harmonic domain model.
Harmonic signal representation in single-phase systems require no extra consideration beyond
the general framework presented in Chapter 2, as their harmonic components are simple phasors
with no sequence-dependent behaviour. Accordingly, it is possible to construct a three-phase
harmonic domain model comprised of three single-phase networks interconnected appropriately,
as in [Rico et al. 2003]. This can be improved upon, for the same reasons that motivated the
development of the method of decomposition to symmetric components. One significant motivation for developing sequence-dependent models is the potential for reduction in the size and thus
4.4
27
HARMONIC DOMAIN VSC MODEL
computational load of the model, which can be realised by taking advantage of analytical results
in the symmetrical framework. As a most immediate example, in a basic three-wire system, no
path is available for zero-sequence current, and consequently, power flows (caveats apply, such
as shared dc-side connections [Pöllänen and Others 2003]). This means that the ac harmonic
state variables of order 0 + 3n may be eliminated without any loss of descriptive power of the
model.
This thesis aims to advance a coherent treatment of three-phase harmonic domain modelling
suitable for use in dynamic (i.e. HSS) models, through some simplifying assumptions about the
system. The first of these is by equating the ac harmonic domain vector Γ(t)Xψ with a vector of
balanced harmonic phasors. For a component of a harmonic phasor to be balanced is equivalent
to it being equal over all three phases by a phase shift defined by the sequence of fundamental
frequency energisation, as represented by the three-phase shift FTM of 2.11.
X ψ = ΨX a
(4.4)
Under this balanced phasor construction, a single harmonic vector, nominally the a phase for
simplicity, is able to simultaneously represent all three phases of an ac quantity such as the
voltage applied to one side of a three-phase inductance, or the set of state variables for the
currents flowing in it. It will be seen in Section 4.4.6.3 that this simplification does not preclude
the modelling of imbalanced components (or other non-characteristic signals).
Secondly, the ac network itself is assumed to be balanced, that is that the impedances on each
phase are assumed to be identical. This property is required for the balance of three-phase
signals to be reflexive across transformation by the network impedances.
The third assumption is the steady state balance of the individual phase switching functions
sψ (t), in the same way as the ac harmonic domain vector was defined to be, allowing all of
the phase switching functions to be described as phase-shifted variants of the phase a switching
~ in a fundamenfunction sa (t). This assumption is concordant with a single control vector M
tal frequency positive-sequence reference frame. This covers a subset of possible vector control
schemes, but does not include control schemes which control the phases individually (and thus
implicitly allow negative sequence control), or schemes which include a negative-sequence fundamental reference frame and associated control vector such as in [Etxeberria Otadui et al. 2007]
to allow direct vector control over unbalanced fundamental power flows.
With these assumptions in place, the ac and dc characteristic harmonic sets of balanced phasors
can be derived. Using the definition of the balanced switching function, the general form of the
harmonic-domain signal over all phases for ψa = 0, ψb =
−2π
3
and ψc =
2π
3
may be written as:
28
CHAPTER 4
THREE PHASE VSC MODELLING
xψ (t) = Γ(t)X ψ
(4.5)
xψ (t) = Γ(t)ΨX a
X
xψ (t) =
Xk ej(kω0 t+kψ)
k∈Z
The following identity holds over all ψ, allowing harmonic components of the same set to be
modulated by terms of order ej3n while maintaining the consistent kψ phase relation to the
fundamental component, which is the defining feature of the balanced set:
(m + 3n)ψ = mψ
(4.6)
Because all switching functions in the network possess half-wave symmetry, then all even ac
terms may be eliminated, giving the more sparse definition:
(m + 6n)ψ = mψ
(4.7)
By applying this definition to the two conjugate elements X+1 and X−1 which together define
the fundamental ac component, as well as to the base dc element X0 , three sets of characteristic
harmonics are defined, which are sufficient to describe all ac and dc signals in a system obeying
the assumptions detailed earlier:





..
..
..
 . 
 . 
 . 






X−6 
X−7 
X−5 











{X+1 } = 
X+1  , {X−1 } = X−1  , {X0 } = X+0 






X+6 
X+5 
X+7 






..
..
..
.
.
.

(4.8)
These characteristic harmonic sets will be used in Section 4.4.3 to enable the description of a
balanced three-phase modulator as a combination of two conjugate transfers, operating linearly
with respect to frequency.
4.4
29
HARMONIC DOMAIN VSC MODEL
4.4.2 Numerical Switching Function Model
At the core of the VSC model of this thesis is the numerical switching function model. This
~ and carrier frequency
maps a given converter operating point, specified as a modulation vector M
multiple mf to a switching function sa (t) and associated spectrum S a .
This is achieved through the direct numerical solution of the switching function, evaluated
discretely at N points. The carrier signal is a symmetrical sawtooth, centred at the origin of
the d-axis.
2π
)
N
(4.9)
2π
2π
) − Mq sin(n )
N
N
(4.10)
vcarrier [n] = saw(mf n
The reference signal is generated as
vref −a [n] = Md cos(n
~ = Md + jMq , in the fixed global synchronous reference frame defined in Section
where M
4.3.2. The switching function is then computed discretely through a Boolean operation on the
magnitude of the two signals as
sa [n] = 2(vref −a [n] > vcarrier [n]) − 1
(4.11)
The discrete approximation to the true switching spectrum is then computed as the FFT (Fast
Fourier Transform) of the discrete switching function as
S a = F{Sa }
(4.12)
and truncated to k = ±2h, twice the maximum harmonic order in the system, the requirement
for population of the truncated Toeplitz FTMs as per equation 2.15.
The direct numerical solution of the switching function is used because of its generality and
suitability for use in computer modelling. Arbitrary accuracy can be achieved with a sufficiently
large choice of N , and the specific choice of carrier signal or alternate modulation method as per
4.3.1.2 can be easily achieved through modification to the reference signals. Analytical solutions
of varying power and generality are available [Holmes 1998], but cannot match the flexibility
afforded by the numerical method in this environment. The question of the sensitivity to the
parameter N is discussed in Section 5.2.3.
30
CHAPTER 4
THREE PHASE VSC MODELLING
This process for generating the numerical switching function is illustrated in Fig. 4.1. The
switching spectrum of Fig. 4.2 is shown for just the characteristic harmonics which will form the
FTMs. It demonstrates the roll-off skew property described in [Sandberg et al. 2005] allowing
for the arbitrarily close modelling of the switching action through an FTM generated from this
spectrum.
Switching Function Generation for mf=9, M=0.8−j0.3
1
0.8
0.6
Magnitude (p.u.)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
Figure 4.1
50
100
150
200
Phase (degrees)
250
300
350
Numerical switching function generation showing the reference, carrier, and switching signals.
4.4.3 Derivation of Three-Phase Transfers
In this section, the frequency transfer matrices for the voltage and current transfers are derived
from their time-domain relations. Although the general procedure for the translation of a generic
LTP system to the harmonic state space is given by the HSS equations of Section 3.2.5, these
are insufficient on their own for the description of the three-phase VSC in the reduced-order
balanced harmonic phasor framework. Instead, the transfers will be developed using specific
results from the principles of the balanced modulator.
The VSC is described by two sets of transfers; from the dc-side voltage to the ac-side voltage,
and from the ac-side current to the dc-side current. These transfers are unrelated to each other
except as they are described by the same patterns of conduction and thus switching functions.
Both of these transfers are stateless, and are therefore described by a static D̄ matrix in the
HSS. The derivation of these transfers proceeds as for a harmonic domain formulation with no
4.4
31
HARMONIC DOMAIN VSC MODEL
Switching Spectrum for mf=9, M=0.8−j0.3
Absolute Magnitude (p.u.)
0.5
0.4
0.3
0.2
0.1
0
−95−89−83−77−71−65−59−53−47−41−35−29−23−17−11 −5 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
Harmonic Order
200
Phase (degrees)
150
100
50
0
−50
−100
−150
Figure 4.2
teristic.
−95−89−83−77−71−65−59−53−47−41−35−29−23−17−11 −5 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
Harmonic Order
Spectrum of the switching function for the characteristic harmonics, exhibiting the roll-off charac-
dynamically varying components, but it will be seen that because of the statelessness of the
modulator, the harmonic domain description is equivalent to the HSS model.
The phase voltages vψ , defined with reference to the dc-side mid-point, may be written generally
as the time-domain products of the individual phase switching functions sψ and the dc bus
voltage (also relative to the dc-side mid-point), which have the harmonic domain formulations:
vψ (t) = sψ (t)
sψ (t) =
X
vdc (t)
2
(4.13)
Sψ,k ejkω0 t
(4.14)
Vdc,m ejmω0 t
(4.15)
k∈Z
vdc (t) =
X
m∈Z
The phase switching functions, because of the balanced condition of the modulator, may be
described as time-shifted variations of the reference phase a switching function sa (t), and
equivalently in the harmonic domain as a harmonic-dependent (and thus implicitly sequencedependent) phase shift of kψ, where ψa = 0, ψb =
−2π
3
and ψc =
2π
3 ,
in accordance with the
32
CHAPTER 4
THREE PHASE VSC MODELLING
derivation of the three-phase shift FTM of equation 2.11:
sψ (t) = Γ(t)S ψ
(4.16)
sψ (t) = Γ(t)ΨS a
X
sψ (t) =
Sa,k ej(kω0 t+kψ)
(4.17)
k∈Z
This is now the fully general equation of the dc voltage to phase voltage transfer in the form of
the harmonic domain product of equation 2.14, which may be expanded to yield the exponents
of the resulting harmonic domain phase voltages:
1
vψ (t) =
2
X
!
X
j(kω0 t+kψ)
Sa,k e
k,m∈Z
jmω0 t
Vdc,m e
(4.18)
1 X
Sa,k ej(kω0 t+kψ) Vdc,m ejmω0 t
2
(4.19)
m∈Z
k∈Z
Va,k+m ej((k+m)ω0 t+kψ) =
!
X
k,m∈Z
With the restriction, assumed for now, of the dc harmonic vector to terms of order m = 0+6p, p ∈
Z, the substitution (k+m)ψ for kψ is performed, as (1±3n)ψ = ψ over all phases. This preserves
the ability of the other phases to be described by the phase a harmonic vector in combination
with the same phase shift, dependent only on the harmonic order n:
vψ (t) =
X
n∈Z
Va,n ej(nω0 t+nψ) =
1 X
Sa,k ej(kω0 t+kψ) Vdc,m ejmω0 t
2
(4.20)
k,m∈Z
The establishment of this condition for the balanced identity of the phases allows for the construction of the equivalent vectorial expression, using the phase-dependent shift matrix of equation
2.11, which gives the form of the harmonic transfer when reduced:
1
Γ(t)Vψ = Γ(t)T {S ψ }V dc
2
1
Γ(t)ΨV a = Γ(t)ΨT {S a }V dc
2
1
V a = T {S a }V dc
2
(4.21)
(4.22)
This transfer could be implemented using a single sparse Toeplitz FTM, where the inputs and
4.4
33
HARMONIC DOMAIN VSC MODEL
outputs consist of full-rank harmonic vectors, giving a square FTM of dimension 2h containing
4h2 terms, where h is the highest harmonic order of the model. It is also possible to partition
this transfer using two conjugate networks, the characteristic harmonic sets 1 + 6p and −1 + 6p,
giving two FTMs of dimension
2h
6
containing
h2
9
terms, an 18-fold reduction in the number
of complex multiplications required, and therefore a 72-fold reduction in the number of scalar
multiplications required when using the tensor representation of complex values described in
Section 4.4.4, and a 216-fold reduction when compared to a full-rank transfer for each phase.


..
..
 .
 . 



Sa,+1 Sa,−5 Sa,−11

Va,−5 



Va,+1  = 1 
Sa,+7 Sa,+1 Sa,−5
 2





Va,+7 
Sa,+13 Sa,+7 Sa,+1



..
.




..
..
.

 .




Sa,−1 Sa,−7 Sa,−13
Va,−7 


 1
Va,−1  = 
Sa,+5 Sa,−1 Sa,−7

 2



Va,+5 

Sa,+11 Sa,+5 Sa,−1



..
.
..
.





 Vdc,−6 


 Vdc,+0 




 Vdc,+6 


..
..
.
.


..
.
(4.23)





 Vdc,−6 


 Vdc,+0 




 Vdc,+6 


..
..
.
.
(4.24)
Such a partitioning also makes explicit the sequence-dependent frequency shifting behaviour of
the modulator. It results in two complex ac networks, each the complex conjugate of the other
in both state-space equations and signals, which combine to provide the necessarily real-valued
signals for real-valued inputs.
A similar transfer can now be derived, from the phase currents to the dc-side current, which
may be written as the sum of the phase currents multiplied by their respective conduction
functions, which give dc currents idc−ψ as the contribution from each phase. These currents can
be physically mapped to the upper part of each converter leg, flowing nominally into the common
positive rail, in accordance with the passive sign convention adopted for the phase currents.
idc (t) =
X
idc−ψ (t)
(4.25)
ψ∈a,b,c
When mapping the phase currents to their contributions to the dc current, the appropriate
transformation is the conduction function of the upper-leg switches cψ (t), which may be defined
relative to the previously defined switching function as cψ (t) =
sψ (t)
2
+ 12 . The constant term is
neglected when constructing the transfer, as both positive and negative sequence phase currents,
34
CHAPTER 4
THREE PHASE VSC MODELLING
when multiplied by this constant, will cancel on the dc side. Because of the assumption of a threewire system, no zero-sequence current which would result in a net contribution to dc current
will be present. To construct a similar model which does allow for zero-sequence flows, due to a
four-wire system or dc-side interconnections, the transfer would necessarily have to be based on
the conduction function including the constant term, and also disallowing the FTM reduction
techniques described earlier. With this assumption present, the transfer in time-domain and
harmonic-domain forms is therefore:
idc−ψ (t) =
idc (t) =
sψ (t)
iψ (t)
2
(4.26)
1 X
sψ (t)iψ (t)
2
(4.27)
ψ∈a,b,c
Because of the linearity and balance of the ac network, if the phase voltages share the balanced
condition of the phase switching functions, it can be concluded that the phase currents are
similarly balanced, and the harmonic domain forms of both the switching function (as per
equation 4.17) and current may be written as harmonic-dependent variations of their respective
phase a harmonic series:
iψ (t) = Γ(t)I ψ
(4.28)
iψ (t) = Γ(t)ΨI a
X
iψ (t) =
Ia,m ej(mω0 t+mψ)
(4.29)
m∈Z
1 X
idc (t) =
2
ψ∈a,b,c
!
X
Sa,k ej(kω0 t+kψ)
!
X
Ia,m ej(mω0 t+mψ)
(4.30)
m∈Z
k∈Z
which when expanded into the form of the dc current sub-component harmonic vectors, provides
the general relation of the exponents:

X

X

ψ∈a,b,c
k,m∈Z


X
X
1

Idc−a,k+m ej((k+m)ω0 t+(k+m)ψ)  =
Sa,k ej(kω0 t+kψ) Ia,m ej(mω0 t+mψ) 
2
ψ∈a,b,c
k,m∈Z
(4.31)
If the harmonic indices of the input harmonic vectors Sa and Ia are restricted to that of our two
4.4
35
HARMONIC DOMAIN VSC MODEL
partioned networks, i.e. the non-zero terms k = ±1+6p and m = ±1+6q, their harmonic domain
products may be divided into two classes; the exponential sum terms where k + m = ±2 + 6r,
and the exponential difference terms where k + m = 0 + 6r.
!
X
X
ψ∈a,b,c
k+m=6r
idc (t) =
Idc−a,k+m ej((k+m)ω0 t+(k+m)ψ) +
X
Idc−a,k+m ej((k+m)ω0 t+(k+m)ψ)
k+m=±2+6r
(4.32)
A simple relation holds for the sum terms, in which the phase shifted exponentials cancel over
the summed phases. The associated terms can be physically interpreted as harmonic currents
which circulate in the converter bridge, but which do not contribute to the net dc current.
X
e(±2+6r)ψ =
ψ∈a,b,c
X
e(±2)ψ = e0 + e±
2π
3
+ e∓
2π
3
=0
(4.33)
ψ∈a,b,c
The result is that the exponential sum terms can be removed from the harmonic domain product,
leaving only difference terms k + m = 0 + 6r, and therefore justifying, through the linearity of
the dc network, the initial assumption of the restricted dc voltage harmonic set. Recalling the
identity 3nψ = 0, it is clear that nψ = 0 for all n = 6p, which is to say that the virtual dc current
contributions from each phase idc−ψ are identical, and so the sum over the phases reduces to a
multiplication by three:
!
X
X
ψ∈a,b,c
n∈Z
Idc−a,n ej(nω0 t+nψ)


X
X
1

=
Sa,k ej(kω0 t+kψ) Ia,m ej(mω0 t+mψ) 
2
ψ∈a,b,c
!
idc (t) = 3
X
n∈Z
Idc−a,n ejnω0 t
(4.34)
k,m∈Z


3 X
=
Sa,k ej(kω0 t+kψ) Ia,m ej(mω0 t+mψ) 
2
(4.35)
k,m∈Z
The equivalent construction of this transfer in vector form, making use of the results for the
selective cancellation of harmonic products based on the restricted indices, can be expressed in
the following form, where A∗ denotes the complex conjugate of A :
36
CHAPTER 4
Γ(t)I dc =
THREE PHASE VSC MODELLING
1 X
Γ(t)T {S ψ }I ψ
2
(4.36)
1 X
Γ(t)T {ΨS a }Ψ I a
2
(4.37)
ψ∈a,b,c
Γ(t)I dc =
ψ∈a,b,c
Γ(t)I dc
I dc
3
= Γ(t)T {S ∗a }I a
2
3
= T {S ∗a }I a
2
(4.38)
(4.39)
With the ac current input vectors in the form of the partitioned conjugate networks, the familiar
Toeplitz structure emerges from the k +m = 6r condition. The conjugate ac networks recombine
on the dc side, as their respective contributions to the dc current add according to their nominal
harmonic index to give the real dc signal, which then necessarily has a purely real zero-frequency
component, and conjugate harmonic terms:

..
.





Idc,−6 


Idc,+0  =




Idc,+6 


..
.

..
 .

Sa,−1 Sa,−7 Sa,−13

3

Sa,+5 Sa,−1 Sa,−7
2


Sa,+11 Sa,+5 Sa,−1

..
 .

Sa,+1 Sa,−5 Sa,−11

3

Sa,+7 Sa,+1 Sa,−5
2


Sa,+13 Sa,+7 Sa,+1



..
 . 


 Ia,−5 


 Ia,+1  +




 Ia,+7 


..
..
.
.


..
 . 


 Ia,−7 


 Ia,−1 




 Ia,+5 


..
..
.
.

(4.40)
With this formulation, the self-consistency of the characteristic harmonic set under the balanced
phasor interpretation of the harmonic domain signals has been demonstrated, while maintaining
the dynamic generality of the model. By partitioning the system into conjugate networks which
are modulated linearly, the sequence-dependent frequency folding behaviour of earlier harmonic
domain models is achieved, but without the need for convoluted transforms, which allows the
state-space description and simulation output to be more legible and intuitive. This shows
the simplifying power of the central assumption of the balanced modulator and associated ac
network.
4.4
37
HARMONIC DOMAIN VSC MODEL
4.4.4 Implementation of LTI Model
The HSS model is implemented in MATLAB as a real-valued LTI state-space system. This is
constructed using a modular approach, with state-space sub-blocks being connected by shared
inputs and outputs using the MATLAB ‘connect’ command, as illustrated in Fig. 4.3. Though
not illustrated, the ac network in itself is comprised of series RL and RC sections, themselves
split into the conjugate characteristic harmonic sets identified in Section 4.4.3. Each sub-block is
a self-contained state-space system, formed from the Ā, B̄, C̄ and D̄ matrices of the transformed
system using the LTP to HSS methods described in Chapters 2 and 3.
Grid Energisation
AC Network
Uncontrolled Transfers
V dc
Va
V sys
I dc
Ia
Figure 4.3
DC Network
Block diagram of the state-space structure of the uncontrolled model.
An additional step is required to form real-valued matrices from the original complex ones. This
is done by separating the real and imaginary parts of the signal, and using a 2x2 tensor to
represent each complex coefficient. A complex multiplication x = yz, x, y, z ∈ C is represented
using real signals as:
"
<x
=x
#
"
=
<y −=y
=y
<y
#" #
<z
=z
(4.41)
When the state-space blocks are joined, the aggregate system is defined by identifying which
signals are to form the inputs and outputs. All signals that are to be stored for processing are set
as outputs. For the uncontrolled system, just the grid side of the ac voltage network is exposed
as an input, giving the form of the input vector as the two conjugate grid voltage vectors:
38
CHAPTER 4

..
.
THREE PHASE VSC MODELLING





Vgrid,−5 





Vgrid,+1 





 Vgrid,+7 


..




.




U =

..


 . 


Vgrid,−7 




V

 grid,−1 




Vgrid,+5 



..
.
(4.42)
4.4.5 Time-Domain Signal Reconstruction
The LTI implementation of the HSS model described here produces sets of harmonic vectors, one
at each solution time-step, as its output. These harmonic output vectors are processed into timedomain signals by multiplying the harmonic vector at each time-step by the vector of complex
harmonic exponentials Γ(t). Theoretically, no distinction is made between a generic full-rank
harmonic vector, and the same vector partitioned into the two characteristic harmonic sets, but
for clarity the process for reconstructing the time-domain a-phase voltage signal is presented in
equation 4.44, directly as it appears in the model. The two vectors of the characteristic harmonic
sets along with their respective Γ(t) vectors being complex conjugates of each other, add to form
a single purely real time series.
h
va (t) = . . . e−j5ω0 t ejω0 t ej7ω0 t
va (t) = Γ(t)V a (t)
(4.43)




..
..
.
.








V
V


h
i
i
 a,−7 
 a,−5 
−j7ω
t
−jω
t
j5ω
t



0
0
0
e
e
. . . Va,−1 
. . . Va,+1  (t)+ . . . e
 (t)




Va,+5 
Va,+7 




..
..
.
.
(4.44)
This reconstruction for ac quantities is specific to the a phase. To generate the time-domain
series for other phases, it is necessary to apply the harmonic-dependent phase shifts of equation
2.11, which recovers the time series of any phase for dynamically varying signals.
4.4
39
HARMONIC DOMAIN VSC MODEL
vψ (t) = Γ(t)ΨV a (t)
(4.45)
As the output an of LTI model, the EMP-domain signals evolve continuously (and in the degenerate steady-state case, are static), and therefore may be linearly interpolated before reconstructing
the time-domain signal without penalty in a way that the time-domain output could not. This
allows the time-domain reconstruction of the HSS simulation carried out at a time-step of 100 µs
to match the output of the EMTDC simulation at a time-step of 10 µs.
4.4.6 Verification of Individual Transfers
To validate the converter model, the individual voltage and current transfers described in Section
4.4.3 are characterised in isolation, without the feedback from the coupling ac and dc networks.
These transfers demonstrate the accuracy of the numerical model used, and illustrate the characteristic of the VSC as a linear frequency modulator.
4.4.6.1
PSCAD Model
The benchmark for the accuracy of the HSS model is an equivalent PSCAD/EMTDC testbench,
constructed of ideal components and with an identical carrier-based PWM scheme. The system
showing the relevant measurement locations is illustrated in Fig. 4.4. So as to match the
assumptions of the HSS model, the PSCAD model derives its phase reference for both the
triangular carrier and the synchronous reference frame from a PLL attached to a fixed threephase source, separated from the active grid to provide a constant phase reference irrespective
of the conditions of the test network. Fixed current and voltage sources provided the test inputs
for the steady-state transfers.
4.4.6.2
Steady-State Transfers
The following results were obtained with the converter operating point defined by the dq-frame
modulation vector of Md = 0.8, Mq = 0.2, a switching frequency of 450 Hz (mf = 9), and
using test inputs of 10 kA rms (leading the reference frame by 90 degrees) and 10 kV dc respectively. The HSS model was energised with the static harmonic vector inputs Vdc,+0 = 10000,
√
√
Ia,+1 = −j5000 2 and Ia,−1 = +j5000 2, resulting in a set of static harmonic vector outputs
in accordance with the stateless input-output characteristic of the converter. These static harmonic outputs were converted to time-domain series for display, and then the same FFT was
applied to the EMTDC and converted MATLAB time series.
40
CHAPTER 4
THREE PHASE VSC MODELLING
idc (t)
Vtest
−
+
+
va (t)
Itest
−
Figure 4.4
PSCAD testbench for individual transfers.
Fig. 4.5 displays the voltage transfer from a zero-frequency dc component only; the resultant ac
voltage is simply the phase a switching function itself, less the common-mode elements, and is
therefore visible as the phase-to-neutral component. This transfer, simulated with a maximum
harmonic order of 157 in order to give clear visual time-domain agreement, nevertheless displays
the Gibbs phenomenon near its multiple periodic jump discontinuities.
The harmonic-domain comparison of the phase a voltage transfer, Fig. 4.6, is indexed linearly
according to the characteristic harmonic set 1 ± 6n, to display only the extant harmonics in
the signal. The full FFT of this real signal necessarily contains energy at positive and negative
frequencies for all present harmonics, but these are not shown in preference to the linear index.
As seen in Fig. 4.8, a small amount of energy at non-characteristic harmonic frequencies is
present in the EMTDC data, which based on the harmonic model developed in this chapter
should not be present in the idealised system; this may be due to numerical error in the control
system. The harmonic and time-domain analyses show otherwise clear agreement.
41
HARMONIC DOMAIN VSC MODEL
AC Voltage from DC
8
HSS
EMTDC
6
4
Voltage (kV)
2
0
−2
−4
−6
−8
0
2
4
6
8
10
Time (ms)
12
14
16
18
20
Time-domain comparison of ac voltage transfer from a constant dc bus voltage.
Figure 4.5
Harmonic Magnitude
2.5
EMTDC
HSS
Voltage (kV)
2
1.5
1
0.5
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
19
25
31
37
43
49
13
19
25
31
37
43
49
Harmonic Angle
400
Phase (degrees)
4.4
300
200
100
0
−47
Figure 4.6
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
Harmonic comparison of ac voltage transfer from a constant dc bus voltage.
42
CHAPTER 4
THREE PHASE VSC MODELLING
DC Current from 50 Hz AC
15
HSS
EMTDC
10
Current (kA)
5
0
−5
−10
0
Figure 4.7
2
4
6
8
10
Time (ms)
12
14
16
18
20
Time-domain comparison of dc current transfer from a positive sequence ac current.
Harmonic Magnitude
2.5
EMTDC
HSS
Current (kA)
2
1.5
1
0.5
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Harmonic Order
Harmonic Angle
350
Phase (degrees)
300
250
200
150
100
50
0
0
2
4
6
Figure 4.8
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Harmonic Order
Harmonic comparison of dc current, all even terms.
4.4
HARMONIC DOMAIN VSC MODEL
4.4.6.3
43
Dynamic Transfers
As described in Chapter 3, the dynamically time-varying form of the general EMP signal allows
the representation of non-characteristic harmonics through the modulation of, in theory, any of
the available harmonic reference frames. This method is used here to extend the reduced-order
converter model to include non-characteristic harmonic signals, principally for the purpose of
modelling the effects of fundamental frequency system voltage imbalances, an important network
condition to study. To create this non-characteristic signal, the fundamental frequency current
input was modulated at −2ω0 , in effect reversing the rotation of the positive sequence phasor
associated with the fundamental frequency harmonic reference. This is encoded here using the
√
√
complex inputs Ia,+1 = +j5000 2e−j2ω0 t and Ia,−1 = −j5000 2ej2ω0 t
As an LTI model, the principle of superposition is obeyed, and using this method, any linear
combination of positive and negative sequence fundamental components could be created using
a superposition of constant and modulated complex inputs, respectively. Additionally, by the
non-uniqueness of the EMP domain, the non-characteristic signal could be generated from a
modulated input in any reference frame, with the strict Toeplitz structure of the transfer matrix
ensuring that the time-domain transfer is equivalent. This characteristic of linearity with respect
to frequency demonstrates how the validity of the transfer for general dynamically varying signals
guarantees the ability to model the transfer of non-characteristic components.
By adopting this convention of a reduced model containing only the characteristic harmonic set of
a balanced system, a sharp distinction is drawn between the steady state of the state-space model,
which must necessarily be balanced, and the dynamic excursions of the state-space model which
may include as a special case non-characteristic harmonics, which are necessarily unbalanced.
In Chapter 5, this distinction is employed when developing the small-signal converter model,
including dynamic control of the converter.
The comparison of negative sequence current transfers in Figs. 4.9 and 4.10 demonstrate the
same high degree of conformity. In the harmonic analysis, the phase components of the nominally
zero 6n terms have been zeroed out so as to not show spurious phase terms.
Figs. 4.11 and 4.12 demonstrate the equivalence of the same signal achieved through the modulation of two different reference frames; the reconstructed signals are exactly identical. In the
7th harmonic reference frame, a negative sequence fundamental signal is generated by the com√
√
plex inputs Ia,+7 = +j5000 2e−j8ω0 t and Ia,−7 = −j5000 2ej8ω0 t . Similarly, by the phasor
interpretation of the balanced harmonic reference frames, to generate the same signal in the
5th harmonic reference frame (associated with a phasor at −5ω0 ), the complex input required
√
√
is Ia,−5 = +j5000 2e+j4ω0 t and Ia,+5 = −j5000 2e−j4ω0 t . It is instructive to note that both
the reference frames (and thus the order of the associated term in the Γ(t) vector used for
time-domain reconstruction) and the frequency complex inputs are separated by 12ω0 .
44
CHAPTER 4
THREE PHASE VSC MODELLING
DC Current from 50 Hz Negative Sequence AC
20
HSS
EMTDC
15
10
Current (kA)
5
0
−5
−10
−15
−20
Figure 4.9
0
2
4
6
8
10
Time (ms)
12
14
16
18
20
Time-domain comparison of transfer from a modulated negative sequence ac current to dc current.
Harmonic Magnitude
5
EMTDC
HSS
Current (kA)
4
3
2
1
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Harmonic Order
Harmonic Angle
Phase (degrees)
400
300
200
100
0
Figure 4.10
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Harmonic Order
Harmonic comparison of transfer from a modulated negative sequence ac current to dc current.
4.4
45
HARMONIC DOMAIN VSC MODEL
DC Current from 50 Hz Negative Sequence AC
20
HSS (−5th)
HSS (7th)
15
10
Current (kA)
5
0
−5
−10
−15
−20
0
2
4
6
8
10
Time (ms)
12
14
16
18
20
Figure 4.11 Time-domain comparison of negative-sequence current transfer by modulated inputs in the -5th
and 7th harmonic reference frames.
Harmonic Magnitude
5
HSS (−5th)
HSS (7th)
Current (kA)
4
3
2
1
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Harmonic Order
Harmonic Angle
Phase (degrees)
400
300
200
100
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Harmonic Order
Figure 4.12 Harmonic comparison of negative-sequence current transfer by modulated inputs in the -5th and
7th harmonic reference frames.
46
CHAPTER 4
THREE PHASE VSC MODELLING
4.5 UNCONTROLLED SYSTEM MODEL
The previous section demonstrated the validity of the uncontrolled harmonic domain VSC model
for steady-state and dynamic inputs. With this functionality satisfied, the VSC model may be
evaluated as a component in a larger HSS model also incorporating passive elements.
The test system, illustrated in Fig. 4.13 consists of a three-phase remote voltage source with an
RL network impedance, interfacing to the VSC through its own RL impedance, with a shunt RC
network providing harmonic filtering. The harmonic response of the filter can be approximately
described by the LC network formed from the filter capacitor and the converter-side inductance,
which eliminates the majority of the harmonic content of the VSC voltage as seen by the grid,
and thus limits harmonic current flows to the grid. The filters, by virtue of being shunt-connected
capacitors, passively supply a small component of reactive power (157 kVA or 0.0785 p.u. at
rated voltage in the simulation to follow) to the network. The dc side of the VSC has a parallel
RC network to model the dc link capacitor, as well as either a small switching loss or alternately
a resistive load. Depending on the chosen system parameters and component values, this basic
structure may be used as a STATCOM, controlled rectifier, or a simplified model of a VSCHVDC terminal.
Figure 4.13
Test network circuit diagram.
The analysis undertaken here uses this system in a STATCOM (Static Synchronous Compensator) configuration, with the VSC acting as a controllable voltage source to deliver a controllable
level of reactive power to the fixed grid. As the simulation is an evaluation of the uncontrolled
model with fixed switching instants, the modulation vector M is chosen in advance such that
the system settles to a steady-state condition where reactive power is being injected into the
grid.
4.5
47
UNCONTROLLED SYSTEM MODEL
4.5.1 STATCOM Operation and Control
In order to place the simulation of STATCOM system dynamics in context, a phasor-based
fundamental-frequency model of the dynamic STATCOM in equilibrium is presented here. This
simplified model illustrated in Fig. 4.14 lacks the shunt filter and dc-side loss component,
compared with the full system. The dynamic STATCOM is distinguished from the conventional
STATCOM by having direct control over the magnitude and angle of the synthesized voltage,
contrasted with power angle control only.
Vconv
Vgrid
Zsys
Vdc
Figure 4.14 Single-line diagram of simplified network, without shunt filter or dc-side loss for the analysis of the
VSC system at equilibrium.
In the steady state, the dc-side capacitor voltage is constant, implying that the real power flow
across the converter is near-zero, with a small component to supply the shunt resistance loss,
which is neglected here. Assuming a constant Md component, manipulation of the Mq component
controls the power angle of the synthesised converter voltage against the remote source. This in
turn sets the equilibrium of dc-side capacitor voltage achieved at zero real power flow through the
converter (and therefore a converter current orthogonal to the converter voltage), as visualised
in Fig. 4.15. In this manner, control of the power angle sets the level of reactive power flow
through the dc voltage.
This equilibrium may be interpreted geometrically as the intersection of a phasor of variable
length following the fixed power angle versus the grid voltage plus the voltage drop along the
interconnecting impedance following the slope of the line X/R ratio (the line impedance angle
θz ). This geometric interpretation allows intuition about the relative passive stability of the
uncontrolled system for different X/R ratios, with respect to changing power angles between the
grid and the synthetic voltage source of the VSC.
Generalising this analysis to systems not in equilibrium illustrates the operational flexibility
of the dynamic STATCOM made possible by the decoupled vector control of the d- and q-axis
voltage components, compared to the STATCOM with power angle control only. Fig. 4.16 shows
the same system after the Md voltage component has been reduced, but before the system can
re-equilibriate. This allows the reactive power flow to be reduced immediately, without the delay
required for the dc voltage to adjust as for a conventional STATCOM. The system current and
thus apparent power flow into the converter then contains a real component, which in time would
drive a new equilibrium at the new, slightly larger power angle. To stabilise the reactive power
48
CHAPTER 4
THREE PHASE VSC MODELLING
flow at this present level, the power angle could be reduced to the point of zero real power flow.
It should be noted that in terms of the power angle, the control action required to satisfy the
equilibrium is contra the short-term action. Because of the two degrees of freedom available for
control, the same equilibrium can be achieved for any absolute value of the modulation vector
M , as the dc voltage will adjust accordingly. The absolute magnitude of the modulation vector
to be maintained at the equilibrium level will therefore be determined by the constraints of the
maximum capacitor voltage (and thus minimum modulation vector magnitude), and the output
voltage held in reserve to provide an immediate increase in reactive power output [Sao et al. 2002].
The STATCOM-based simulations in this chapter therefore use a baseline modulation vector of
0.8.
q-axis
Isys
Isys Zsys
θz
Vgrid
d-axis
δ
Vconv = M.Vdc
Figure 4.15
Phasor diagram showing VSC system at equilibrium, supplying reactive power.
q-axis
Isys
Isys Zsys
θz
Vgrid
d-axis
δ
Vconv = M.Vdc
Figure 4.16 Phasor diagram showing VSC system not at equilibrium after adjusting modulation vector, reducing reactive power supply.
4.5
49
UNCONTROLLED SYSTEM MODEL
4.5.2 Uncontrolled System Verification
As for the verification of the individual converter transfers, the benchmark for the evaluation
of the uncontrolled system HSS model is a PSCAD/EMTDC simulation constructed of ideal
components and with an identical control scheme in an independent synchronous frame. The
HSS model results were produced with a maximum harmonic order of 109. Table 4.1 lists the
parameters for the test system configured to function as a STATCOM supplying reactive power
to the grid.
Table 4.1
Uncontrolled test system parameters.
Parameter
Value
Base voltage
Base MVA
Fundamental frequency
Switching frequency
Modulation vector
Source resistance
Source inductance
Converter resistance
Converter inductance
Filter resistance
Filter capacitance
DC-side capacitance
DC-side resistance
Converter LC corner frequency
Steady-state MVA
Steady-state dc voltage
Vbase = 10 kV
Sbase = 2 MVA
f0 = 50 Hz
fsw = 750 Hz
M = 0.8 − j0.03
Rs = 0.25 Ω
Ls = 0.01 H
Rc = 1 Ω
Lc = 0.04 H
Rf = 0.1 Ω
Cf = 5 µF
Cdc = 300 µF
Rdc = 10 kΩ
fco = 355.881 Hz
Sss = 2.8 MVA
Vss = 29.5 kV
Per-unit basis
15 p.u.
0.005 p.u.
0.0628 p.u.
0.02 p.u.
0.251 p.u.
0.002 p.u.
12.7 p.u.
0.212 p.u.
2000 p.u.
7.12 p.u.
1.4 p.u.
The dc capacitor voltage is a useful measure of the accuracy of the simulation, because it
tracks the integral of the real power flow through the converter (less dc-side losses). In Fig.
4.17, very close agreement of the dc voltage is observed between the EMTDC and HSS models,
demonstrating the ability of the model to closely track the power flow during the start-up
transient from an initial state of all variables zeroed. This measurement allows the observation
of a decaying oscillation close to the fundamental frequency, driven by the initial asymmetry
between the phases in the presence of an increasing dc voltage. Visualising the ac voltage in
Fig. 4.18 similarly demonstrates the dynamic accuracy of the model.
A further test of the validity of the dynamic model of the full system involves perturbing the
model with a negative sequence grid voltage component, achieved through the modulation of a
grid voltage harmonic input. The modulated harmonic voltage signal results in the propagation
of modulated signals through the converter as currents, interacting on the dc-side network to
50
CHAPTER 4
THREE PHASE VSC MODELLING
give a modulated bus voltage, which is then fed back onto the ac network through the voltage
transfer FTM, forming a feedback loop. Correct modelling of the system in the HSS for these
inputs therefore relies on the complete linearity with respect to frequency of all parts of the
model, which is implied by the validity of the dynamic model.
This is achieved using the superposition of constant (giving the positive sequence fundamental frequency phasor) and exponentially modulated (giving the negative sequence fundamental frequency phasor) inputs to the fundamental frequency harmonic input, written here as
√
√
Vgrid,+1 = 5000 √23 (1 + 0.1e−j2ω0 t ) and Vgrid,−1 = 5000 √23 (1 + 0.1e+j2ω0 t ), to give a 10% imbalance which was switched in at t = 250 ms, once the start-up transient had begun to stabilise.
Fig. 4.19 shows the response of the dc voltage to this imbalance, which clearly demonstrates
the accuracy of the response of the model to the imbalance, though showing an already present
small discrepancy in the average level of the dc voltage. The comparison of the dc converter
current in Fig. 4.20 shows that the exponentially modulated signals at all stages of the feedback
loop described above provide meaningful time-domain reconstructions.
4.6 CONCLUSION
The numerical harmonic state space model of the VSC was demonstrated to accurately model
uncontrolled current and voltage transfers, in isolation and as part of a full converter system.
Importantly, it was demonstrated that by constructing the VSC model as a linear modulator
acting on conjugate characteristic harmonics sets, the frequency-coupling action of the VSC
can be captured in full dynamic generality by a reduced-order model. The controlled VSC
model developed in the next chapter will rely on the orthogonality of the numerical vector
control scheme and decoupled three-phase signal representation as implemented here to achieve
an accurate linearised model.
51
CONCLUSION
DC Voltage Startup Transient
12
10
Voltage (kV)
8
6
4
2
0
HSS
EMTDC
−2
0
10
20
30
40
50
Time (ms)
60
70
80
90
100
Transient evolution of dc voltage during STATCOM start-up.
Figure 4.17
AC Voltage Startup Transient
6
4
2
Voltage (kV)
4.6
0
−2
−4
HSS
EMTDC
−6
0
5
Figure 4.18
10
15
20
25
Time (ms)
30
35
40
45
Transient evolution of phase a voltage during STATCOM start-up.
50
52
CHAPTER 4
THREE PHASE VSC MODELLING
DC Voltage with 10% Imbalance at 250 ms
20.5
20
Voltage (kV)
19.5
19
18.5
HSS
EMTDC
18
240
Figure 4.19
250
260
270
Time (ms)
280
290
300
Transient evolution of dc voltage following introduction of 10% imbalance at 250 ms.
DC Current with 10% Imbalance at 250 ms
0.15
0.1
Current (kA)
0.05
0
−0.05
−0.1
−0.15
HSS
EMTDC
−0.2
240
Figure 4.20
250
260
270
Time (ms)
280
290
300
Transient evolution of dc current following introduction of 10% imbalance at 250 ms.
Chapter 5
VSC MODEL WITH CONTROL
5.1 INTRODUCTION
With the development and verification of the VSC model in Chapter 4, the ability of the HSS
to completely capture the dynamics of a power system comprised of linear elements and an
uncontrolled VSC defined by a fixed switching function has been demonstrated. By formulating
the VSC as a linear time-invariant system, the VSC system gains the properties of the LTI statespace described in Section 3.2.1; the system may be solved directly for its steady state solution
for arbitrary inputs, and its dynamics may be evaluated through examination of its eigenvalues.
These properties allow new analyses for uncontrolled VSC systems, however the most important
studies of converter system dynamics require the closed-loop controlled response to be modelled.
This requires the extension of the VSC model to include the response to changes in the control
input. As discussed in Section 5.2, this is a problem of linearisation. In this chapter, it will be
shown how the numerical switching model and synchronous reference frame for the uncontrolled
VSC developed in Chapter 4 provides an ideal basis for extension to a small-signal controlled
HSS model for analysing harmonic interaction.
Controlled state-space models have been developed for various converters, using state-space
averaging techniques. In a precursor to modern HSS formulations, [Möllerstedt and Bernhardsson 2000] presented an HSS model of a controlled locomotive rectifier with idealised switching
based on linearised pre-computed waveforms, able to produce closed-loop stability criteria. A
significant class of averaged state-space model for ac systems are the Dynamic Phasor models, as formulated by Stankovic et al. [Stanković et al. 2000], in which generalised averaging
techniques are applied over a small chosen set of harmonic reference frames to the governing
differential equations of the system. The resulting equations, generally developed for individual
phases, are time-invariant but not required to be linear. This means that more complicated
non-linear time-invariant controllers may be included as in [Deore et al. 2012], and non-linear
system dynamics and measurement techniques may be included without linearisation. To gain
analysis for these models in the s-domain however, requires a secondary linearisation step to be
54
CHAPTER 5
VSC MODEL WITH CONTROL
carried out. This is contrasted with the inherently linear HSS model developed in this chapter
that results from the extension of the uncontrolled harmonic-domain model, and the similarly
linearised implementation of Switching Instant Variation applied to TCR and line-commutated
HVDC systems by Orillaza and Hwang in [Orillaza et al. 2010].
In this chapter, the controlled small-signal model of the VSC is derived through a process of
linearisation of the numerical switching model using the orthogonal dq-axis control structure
developed in Chapter 4, and the limits of the linearisation process are characterised. The
structure of the complete small-signal model is described in Section 5.2.1 as a linear combination
of modular transfers, delineating the large-signal and small-signal modes of the model. Concepts
for the measurement of dynamic harmonic signals for use as control inputs are introduced in
Section 5.4.1, and applied to the measurement and control of reactive power and capacitor
voltage in a simple closed-loop STATCOM control system. The resulting system is validated
against time-domain simulation in PSCAD in Section 5.5, used to demonstrate the relevance of
LTI analysis techniques when applied to the HSS model, and to perform a test of the dependence
on the closed-loop control response to harmonic coupling terms.
5.2 LINEARISATION STRATEGY
This section describes the application of linearisation techniques to the uncontrolled VSC model
of Chapter 4, in order to gain a fully LTI harmonic state-space model of the controlled converter.
Using the harmonic-domain vector notation, equation 5.1 shows the form of the dynamic transfer
to the phase a voltage for a fixed switching function, representative of the dynamic transfer for
all phases in the balanced network.
1
V a (t) = T {S a }V dc (t)
2
(5.1)
The transfer is defined by the switching function sa (t), and equivalently in the harmonic domain,
the Toeplitz matrix formed from its spectrum S a . For fully general control, the switching
function becomes a function of the time-variant control inputs Md (t) and Mq (t).
1
V a (t) = T {S a (Md (t), Mq (t))}V dc (t)
2
(5.2)
This equation, although accurately describing the general controlled transfer, is neither linear
(as the switching function is a non-linear function of the control inputs), nor time-invariant.
5.2
55
LINEARISATION STRATEGY
Formulation of the LTI state-space model therefore requires a dual linearisation around both
the switching function and the converter state variables.
Firstly, the switching function as a function of the control vector input must be linearised around
small changes each of the decoupled control vector components. This requires the construction
of the partial differential switching spectra
∂S a
∂Md
and
∂S a
∂Mq .
The differential switching spectra are
obtained using the numerical switching function model, by computing the switching functions
and associated spectra for the base case control vector, and for the base case control vector
plus a small finite differential control input ∆Md , and then normalising the resulting differential
spectra to the magnitude of the differential control input. Because of the linearity of the FFT,
it does not matter whether the difference taken is between the time-domain switching functions,
or the resultant spectra, although both the description here and implemented model use the
differential spectra.
(S a (Md−base + ∆Md , Mq−base ) − S a (Md−base , Mq−base ))
∂S a
=
∂Md
∆Md
(5.3)
(S a (Md−base , Mq−base + ∆Mq ) − S a (Md−base , Mq−base ))
∂S a
=
∂Mq
∆Mq
(5.4)
The partial differential spectra obtained in 5.3 and 5.4, when substituted into the original dynamic transfer of 5.1, yield the formulation of equation 5.5. The switching spectrum, previously
a non-linear function of the control inputs is now replaced by a linear superposition of the base
case spectrum and the two partial differential switching spectra, scaled by the instantaneous
values of the scalar control inputs Md (t) and Mq (t).
This structure is time-invariant, with the constant-valued linearised frequency transfer matrices
being suitable for embedding in a dynamic model. This structure is not linear however, as the
output is dependent on the multiplication of two dynamic variables, namely the scalar control
inputs Md (t), Mq (t) with the dc voltage harmonic vector V dc (t).
1
V a (t) =
2
∂S a
∂S a
T {S a } + δMd (t)T
+ δMq (t)T
V dc (t)
∂Md
∂Mq
(5.5)
The solution to this is the linearisation of the converter state variables around the base-case
operating point. By assuming that the harmonic state variables remain constant over small
variations in the control inputs or changes in the external energisation applied to the system,
the relation can be reformulated to be completely linear and time-invariant.
56
1
V a (t) =
2
CHAPTER 5
VSC MODEL WITH CONTROL
∂S a
∂S a
T {S a }V dc (t) + δMd (t)T
V dc−base + δMq (t)T
V dc−base (5.6)
∂Md
∂Mq
The harmonic vectors for the base-case state variables are multiplied at build-time by the partial
differential switching FTM, giving the partial differential output vectors:
∂V a
1
∂S a
V dc−base
= T
∂Md
2
∂Md
∂V a
1
∂S a
= T
V dc−base
∂Mq
2
∂Mq
(5.7)
(5.8)
Together with the original large-signal FTM, these partial differential output vectors add to form
the total controlled transfer, giving a single harmonic vector output which may be connected to
a passive ac network.
1
∂V a
∂V a
V a (t) = T {S a }V dc (t) + δMd (t)
+ δMq (t)
2
∂Md
∂Mq
(5.9)
A similar procedure may be applied to the current transfer, resulting in a similar controlled
transfer, linearised around the base case inductor current.
∂I dc
3
∂S ∗a
= T
I a−base
∂Md
2
∂Md
3
∂S ∗a
∂I dc
= T
I a−base
∂Mq
2
∂Mq
3
∂I dc
∂I dc
I dc (t) = T {S ∗a }I a (t) + δMd (t)
+ δMq (t)
2
∂Md
∂Mq
(5.10)
(5.11)
(5.12)
These linearisations correspond to two modes of error as the small-signal model diverges from the
base-case operating point. Firstly, the linearised model of how the switching function changes
as a result of control inputs becomes less accurate with increasing deviation. This contribution
to model error will be discussed in Section 5.2.3, and will be ultimately overshadowed by the
second mode of error. This second mode of error is the result of the fact that any change in
the converter state variables away from the harmonic steady-state of the base case will not be
5.2
LINEARISATION STRATEGY
57
reflected in the base-case harmonic vectors built in to the partial differential transfer vectors.
5.2.1 Small-Signal Model Structure
As described in Chapter 4, the three-phase bridge which forms the kernel of the VSC is a stateless
modulator, capable of indiscriminately coupling voltages and currents through its terminals. It
is the linking of the modulator with the ac-side inductances and dc-side capacitance that lends
these voltages and currents a unilateral inertia, and therefore gives the VSC its structure as a
directional modulator of currents and voltages. The same inertia of these state variables, when
extended to the harmonic state space, provides the basis of the linearisation described above.
This marks the distinction between the uncontrolled model with fixed switching instants, which
is valid for all real signals applied to the system, and the small-signal controlled model, which
is valid only for the region of signal-space surrounding the base case operating point. Because
the linear model obeys the principle of superposition, it is possible to separate the open-loop
model into three orthogonal modes and associated responses, setting aside for the moment the
presence of measurement and control elements:
1. The model with uncontrolled transfers only, energised with the base-case inputs, yielding
the base-case state variables.
2. The model with uncontrolled transfers only, energised with a disturbance to the base-case
inputs (e.g. step change in grid voltage magnitude or angle, or a dynamic negative-sequence
disturbance), yielding the deviation from the base case state variables (which cannot be
incorporated in the linearised transfers).
3. The model with controlled and uncontrolled transfers, energised with externally derived
control inputs, yielding the small-signal deviation from the base case due to control action.
Fig. 5.1 illustrates this linearised converter structure as a harmonic state-space model, showing
how the contribution from the control action is realised as a set of exogenous forcing components.
It is important to note that even in the small-signal model, all energy flow in the system still
passes through the large-signal uncontrolled transfers and passive networks, including the smallsignal component induced in the ac and dc networks and subsequently coupled back to the
converter. This reveals the dual nature of the linearisation error as the converter departs from
the base-case operating point:
1. Deviations in the linked converter state variables are not reflected in the static control
transfers.
58
CHAPTER 5
AC Network
VSC MODEL WITH CONTROL
Uncontrolled Transfers
V dc
Va
DC Network
I dc
Ia
Controlled Transfers
∂V a
∂Md
∂I dc
∂Md
∂V a
∂Mq
∂I dc
∂Mq
δMd
Figure 5.1
δMq
Block diagram of small-signal open-loop control model.
2. The induced small-signal components, once coupled back through the passive networks,
are modulated by the base-case uncontrolled transfer only.
The open-loop model of Fig. 5.1 does not allow for the coupling of the system state variables to
the control inputs, and therefore between the orthogonal modes of the system.
5.2.2 PWM Sampling Error
There exists another mode of error which is directly related to the nature of the switching
converter, and its description in this thesis as an LTI system using the HSS formulation. This
is the problem of sampling, as during the operation of the converter, any updates in the ordered
control vector and thus switching function are only realised at the time of the next switching
instant, giving the closed-loop operation of the PWM converter the characteristic of a continuous
sampled-data system [Fang and Abed 1999]. The linearised model described in Section 5.2
actuates on control inputs instantaneously. The worst case is sampling at 2f0 , as in the case
of thyristor-controlled networks, with a maximum delay of
T
2
before the control signal may be
sampled by an individual phase. This process may be approximated as a fixed delay [Mathur and
Varma 2002], chosen as a midpoint between two possible extremes and assuming sampling can
5.2
59
LINEARISATION STRATEGY
take place on any phase, as in [Orillaza 2011]. This fixed delay may be then approximated as a
rational polynomial in the s-domain through a Padé approximation or similar, introducing a nonminimum phase element[Vajta 2000]. For the PWM-switched converter, the sampling frequency
is 2fc , twice the carrier frequency, with the maximum delay similarly given by
T
2fc .
Because this
effect is less significant at the switching frequencies of PWM converters, it is neglected in this
model.
60
CHAPTER 5
VSC MODEL WITH CONTROL
5.2.3 Switching Function Linearisability
As a first step towards establishing the validity of the linearised control model, the linearisability
of the switching function itself is examined. This is the test of the linearisation achieved using the
partial differential switching spectra of equations 5.3 and 5.4, of which an example is illustrated
in Fig. 5.2. Notably, these spectra do not exhibit the frequency roll-off characteristic associated
with the modelling criterion of Sandberg et al. [Sandberg et al. 2005], as the base switching
functions themselves do.
Linearised Md Spectrum
Absolute Magnitude (p.u.)
0.5
0.4
0.3
0.2
0.1
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
19
25
31
37
43
49
13
19
25
31
37
43
49
Linearised Mq Spectrum
Absolute Magnitude (p.u.)
0.5
0.4
0.3
0.2
0.1
0
Figure 5.2
0.8 − j0.03
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
~ base =
Spectra of linearised switching functions, normalised to unit control inputs, for mf = 9, M
The numerically derived switching function as described in Section 4.4.2 provides a mapping
from any given converter control operating point to an output switching spectrum. The linearisability around an operating point is tested by comparing the spectrum produced from the
linear combination of the base and the partial differential spectra for a control input shown in
equation 5.14, with the one evaluated directly at the new operating point including the control
change (5.15).
5.2
61
LINEARISATION STRATEGY
S a−base = S a (Md−base , Mq−base )
S a−linearised = S a−base + δMd
(5.13)
∂S a
∂S a
+ δMq
∂Md
∂Mq
(5.14)
S a−direct = S a (Md−base + δMd , Mq−base + δMq )
(5.15)
Figs. 5.3, 5.4, and 5.5 show the absolute magnitude of the differences between the individual
harmonic components of these two spectra, which are the switching function linearisation errors.
The absolute value of the linearisation error is considered here, as this relates directly to the
error in the signal introduced to the system through the control action. Only the characteristic
harmonics which comprise the switching FTMs are considered.
~ base = 0.8 − j0.03, with
The results here are produced using a base-case modulation vector of M
a switching frequency of 450 Hz. As seen in the figures, with increasing deviation of the control
signals, the error in the higher harmonic frequencies increases proportionally greater than the
linear trend at the fundamental frequency; this is the result of the non-linearity of the PWM
process itself. The significance of error at higher frequencies is reduced, as the admittance to
these frequencies in the passive ac and dc networks is respectively low and high.
The absolute linearisation error of the fundamental component is noted in Table 5.1. The error
in the fundamental component is crucial, as it contributes to the fundamental frequency power
flow. The trend in the absolute error is linearly related to the magnitude of the control input,
and the errors for changes in both the Md and Mq components are predominantly real, summing
together when actuated simultaneously.
The accuracy that can be obtained depends on the discrete angular step size used for the
numerical switching model detailed in Section 4.4, as the exact switching instant is determined
from a numerical comparison of two discrete series. The results presented here and elsewhere in
this thesis have used a sample count of N = 524 288 or 219 per cycle, determined empirically to
balance computation time with accuracy. A full analysis of the sensitivity of the model to this
parameter is outside the scope of this thesis.
Table 5.1
Fundamental component absolute linearisation error (per-unit basis).
Control input (p.u.)
Error for Md change
Error for Mq change
Error for Md and Mq change
0.01
0.05
0.10
0.20
5.6919 × 10−5
2.6953 × 10−4
5.3990 × 10−4
0.0011
5.6661 × 10−5
2.9359 × 10−4
5.6960 × 10−4
0.0012
1.0862 × 10−4
5.5165 × 10−4
0.0011
0.0022
62
CHAPTER 5
−4
Linearisation Error, Md − 0.01
Absolute Magnitude (p.u.)
x 10
1
0
−47
−41
−35
−29
−23
−17
−3
3.5
Absolute Magnitude (p.u.)
VSC MODEL WITH CONTROL
−11
−5
1
7
Harmonic Order
13
19
25
31
37
43
49
19
25
31
37
43
49
19
25
31
37
43
49
19
25
31
37
43
49
Linearisation Error, Md − 0.05
x 10
3
2.5
2
1.5
1
0.5
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
Linearisation Error, Md − 0.10
Absolute Magnitude (p.u.)
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
Linearisation Error, Md − 0.20
Absolute Magnitude (p.u.)
0.06
0.05
0.04
0.03
0.02
0.01
0
−47
−41
−35
−29
Figure 5.3
−23
−17
−11
−5
1
7
Harmonic Order
13
Absolute linearisation error for changes in Md
63
LINEARISATION STRATEGY
−4
Linearisation Error, Mq − 0.01
Absolute Magnitude (p.u.)
x 10
2
1
0
−47
−41
−35
−29
−23
−17
−3
Absolute Magnitude (p.u.)
3.5
−11
−5
1
7
Harmonic Order
13
19
25
31
37
43
49
19
25
31
37
43
49
19
25
31
37
43
49
19
25
31
37
43
49
Linearisation Error, Mq − 0.05
x 10
3
2.5
2
1.5
1
0.5
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
Linearisation Error, Mq − 0.10
Absolute Magnitude (p.u.)
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
Linearisation Error, Mq − 0.20
0.05
Absolute Magnitude (p.u.)
5.2
0.04
0.03
0.02
0.01
0
−47
−41
−35
−29
Figure 5.4
−23
−17
−11
−5
1
7
Harmonic Order
13
Absolute linearisation error for changes in Mq
64
CHAPTER 5
−4
Absolute Magnitude (p.u.)
6
Linearisation Error, Md − 0.01, Mq − 0.01
x 10
5
4
3
2
1
0
−47
−41
−35
−29
−23
−3
8
Absolute Magnitude (p.u.)
VSC MODEL WITH CONTROL
−17
−11
−5
1
7
Harmonic Order
13
19
25
31
37
43
49
25
31
37
43
49
25
31
37
43
49
25
31
37
43
49
Linearisation Error, Md − 0.05, Mq − 0.05
x 10
6
4
2
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
19
Linearisation Error, Md − 0.10, Mq − 0.10
Absolute Magnitude (p.u.)
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
−47
−41
−35
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
19
Linearisation Error, Md − 0.20, Mq − 0.20
Absolute Magnitude (p.u.)
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
−47
−41
−35
Figure 5.5
−29
−23
−17
−11
−5
1
7
Harmonic Order
13
19
Absolute linearisation error for changes in Md and Mq
5.3
OPEN-LOOP CONTROLLED MODEL
65
5.3 OPEN-LOOP CONTROLLED MODEL
Demonstration of the validity of the linearisation strategy described in Section 5.2, and the
characterisation of the limits of the small-signal model created by that linearisation, require the
modelled transfers to be evaluated as a component in a full power system, allowing deviation
of the state variables away from the harmonic steady state of the base case through coupling to
the passive networks.
The system parameters used are that of the uncontrolled STATCOM system verification presented in Section 4.5.2, and both the MATLAB and PSCAD models are initialised to the periodic
steady-state of the base case. The MATLAB base case was obtained using the direct solution
by inversion of the dynamic matrices detailed in Section 3.2.6, and the PSCAD base case was
established by a simulation of the uninitialised system to a periodic steady state at t = 2 s, and
then using the facility for initialising subsequent simulations with its snapshot.
5.3.1 Step Responses in the Time Domain
Step inputs applied to the δMd and δMq control inputs give an instantaneous change in the converter power angle and magnitude, resulting in a transient evolution to a new steady state. Figs.
5.6 and 5.7 show the response of the converter model to a step change of the input modulation
vector on the direct axis, giving a temporary boost to the direct-axis voltage component, and
with reference to the STATCOM dynamics described in Section 4.5.1, providing a fast-acting
reactive power support response.
As for the uncontrolled model simulations of Section 4.5.2, the dc voltage measurement is used
here to provide a measure of the integral power flow, which is a sensitive indicator of total system
dynamics. The simulations identify steady-state offset errors in the dc voltages, detailed in Fig.
5.8 which increase with the magnitude of the control input (approximately 0.05 kV and 0.2 kV
respectively), however, as identified in Fig. 5.9, the controlled model accurately captures the
oscillatory mode of the transient response.
In Figs. 5.10 and 5.11, both the transient response and the steady-state are modelled accurately
for step δMq inputs of both magnitudes.
66
CHAPTER 5
VSC MODEL WITH CONTROL
DC Voltage Following 0.01 Md Step Change
29.6
29.5
Voltage (kV)
29.4
29.3
29.2
29.1
29
HSS
EMTDC
28.9
0
Figure 5.6
100
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following the application of a step control input of Md = 0.01.
DC Voltage Following 0.05 Md Step Change
30
29.5
Voltage (kV)
29
28.5
28
27.5
HSS
EMTDC
27
0
Figure 5.7
100
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following the application of a step control input of Md = 0.05.
5.3
67
OPEN-LOOP CONTROLLED MODEL
DC Voltage Following 0.05 Md Step Change
27.6
27.55
27.5
Voltage (kV)
27.45
27.4
27.35
27.3
27.25
27.2
HSS
EMTDC
27.15
1980
Figure 5.8
state offset.
1982
1984
1986
1988
1990
Time (ms)
1992
1994
1996
1998
2000
DC bus voltage following the application of a step control input of Md = 0.05, detailing the steady-
DC Voltage Following 0.01 Md Step Change
29.65
29.6
29.55
Voltage (kV)
29.5
29.45
29.4
29.35
29.3
HSS
EMTDC
29.25
100
105
110
115
120
125
Time (ms)
130
135
140
145
150
Figure 5.9 DC bus voltage following the application of a step control input of Md = 0.01, detailing the initial
transient response.
68
CHAPTER 5
VSC MODEL WITH CONTROL
DC Voltage Following 0.01 Mq Step Change
33
32.5
32
Voltage (kV)
31.5
31
30.5
30
29.5
HSS
EMTDC
29
0
100
Figure 5.10
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following the application of a step control input of Mq = 0.01.
DC Voltage Following 0.05 Mq Step Change
46
44
42
Voltage (kV)
40
38
36
34
32
30
HSS
EMTDC
28
0
100
Figure 5.11
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following the application of a step control input of Mq = 0.05.
5.3
69
OPEN-LOOP CONTROLLED MODEL
5.3.2 Step Responses in the State Space
With the basic validity of the control transfers established, the special properties of the harmonic
state space formulation of the VSC provide previously inaccessible insight to the operation of the
converter system. The raw output of the HSS model is available, in the form of the full harmonic
vector outputs for each individual state variable and output node of the state-space subsystems.
This provides the dynamic evolution of every harmonic component of a given signal, with the
advantage that this is obtained without any need for a windowed FFT operation, and the caveat
that the dynamic harmonic-domain signal is non-unique. Rico et al. [Rico et al. 2003] contains
a comparison between measurements of harmonic components achieved through windowed FFT
methods and the HSS, demonstrating how the HSS forms an ideal basis for the extension of
steady-state harmonic measurement indices to their transient evaluation.
The fundamental component of any state variable, which is the least amenable to conventional
Fourier analysis, may be readily extracted in this way. Fig. 5.12 shows the breakdown of the
output voltage into the d− and q−axis components, which identify with the real and imaginary
components of the complex signal in the synchronous reference frame.
−0.42
12
−0.44
11
0
100
200
Figure 5.12
300
400
500
Time (ms)
600
700
800
900
Fundamental components of harmonic state-space output.
−0.46
1000
Quadrature−Axis AC Voltage (kV l−n)
Direct−Axis AC Voltage (kV l−n)
AC Voltage Following 0.05 Md Step Change
13
70
CHAPTER 5
VSC MODEL WITH CONTROL
Under the passive sign convention used for the converter model, positive quadrature-axis converter current indicates a supply of reactive power from the converter to the grid. Fig. 5.13
shows a similar breakdown of the converter current into the orthogonal components of the synchronous dq reference frame, showing the temporary increase in reactive power output. Fig.
5.14 shows the evolution of the dc voltage and current by examination of the zero-frequency
component, making it possible to identify the main resonant modes of the system in the time
domain without interference from the higher-order harmonic components. This ability to extract
the fundamental component in real time will be used when developing the closed-loop control
system.
5.3.3 Pole-Zero Analysis of the Open-Loop System
In the time domain, examination of any of these series reveals a dominant resonant pole-pair at
a frequency of approximately 54 Hz, as well as an overdamped response. This is borne out by
inspection of the pole-zero plot for the open-loop system of Fig. 5.15, in which the dominant
poles are identified in Fig. 5.16, and can be identified with the observed time-domain response
(342 rad s−1 = 54.43 Hz). These oscillatory modes are the product of resonances between the dc
link capacitor and the inductive ac network, as mediated by the action of the converter.
The large-scale structure of the pole-zero plot shows the same poles repeated up and down the
imaginary axis, as the same dynamic response is recreated in multiple harmonic reference frames.
This is revealing of the manner in which higher-order resonances in the system may be coupled
down to a lower frequency range of interest, such as the control bandwidth.
71
OPEN-LOOP CONTROLLED MODEL
0.04
0.3
0.02
0.28
0
0.26
−0.02
0.24
−0.04
0
100
200
Figure 5.13
300
400
500
Time (ms)
600
700
800
900
Quadrature−Axis AC Current (kA)
Direct−Axis AC Current (kA)
AC Current Following 0.05 Md Step Change
0.22
1000
Fundamental components of harmonic state-space output.
0.05
30
0
28
−0.05
0
100
200
Figure 5.14
300
400
500
Time (ms)
600
700
800
900
Fundamental components of harmonic state-space output.
26
1000
DC Voltage (kV)
DC Voltage and Current Following 0.05 Md Step Change
DC Current (kA)
5.3
72
CHAPTER 5
VSC MODEL WITH CONTROL
Pole−Zero Map
4
2.5
x 10
2
1.5
Imaginary Axis (seconds−1)
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−60
−50
−40
−30
−20
−10
Real Axis (seconds−1)
Figure 5.15
Pole-zero map of open-loop controlled system, showing entire system.
0
73
OPEN-LOOP CONTROLLED MODEL
Pole−Zero Map
1000
800
600
System: SYSTEM
Pole : −19.4 + 341i
Damping: 0.0568
Overshoot (%): 83.6
Frequency (rad/s): 342
400
Imaginary Axis (seconds−1)
5.3
200
System: SYSTEM
Pole : −3.53
Damping: 1
Overshoot (%): 0
Frequency (rad/s): 3.53
0
System: SYSTEM
Pole : −19.4 − 341i
Damping: 0.0568
Overshoot (%): 83.6
Frequency (rad/s): 342
−200
−400
−600
−800
−1000
−50
−40
−30
−20
−10
0
Real Axis (seconds−1)
Figure 5.16
Pole-zero map of open-loop controlled system, highlighting dominant poles.
74
CHAPTER 5
VSC MODEL WITH CONTROL
5.4 CLOSED-LOOP CONTROLLED MODEL CONSTRUCTION
The controlled HSS model of the VSC developed in this chapter is an LTI state-space system,
accepting two scalar state-space signals as control inputs. This section describes the method
for the measurement of the system quantities to be controlled, and the formulation of a simple
control system within the state space. This will yield a state-space model of the complete
closed-loop converter system including full harmonic coupling effects, and with this, access to
LTI analysis techniques. The controlled model is validated for perturbations to the grid and
control setpoint, and used to generate results for control analysis and harmonic coupling effects.
5.4.1 Control Variable Measurement
To gain a measurement of a signal in the state-space model, the properties of the HSS signal
representation must be considered. Because of the presence of multiple harmonic reference
frames, the dynamic EMP signal is non-unique, in contrast to the unique periodic steady-state
as defined by the balanced characteristic harmonic phasor set, which the EMP signal degenerates
to under steady-state harmonic energisation, in accordance with the transient relaxation property
described in Section 3.2.
This property of relaxation may be exploited to gain a measurement of any particular harmonic
component of an EMP signal in a stable HSS system, by assuming that the dynamic harmonic
signal undergoes a similar continuous relaxation towards a quasi-steady state condition in which
signal energy is concentrated in the lowest-order harmonic reference frames. This yields slowlyvarying harmonic components which may be used as measurement of individual harmonics as
would be derived through windowed FFT methods, though available in real time, without the
problems associated with an actual window.
The specific measurements discussed below are the dc voltage and reactive power. Together,
these variables can uniquely specify the operating point of the single-converter system, and act
as control variables for converter configurations such as the STATCOM-type system used in this
chapter, as well as such others as controlled rectifier systems where the goal is to provide a fixed
dc output voltage and minimal reactive power draw.
5.4.1.1
DC Voltage
In a STATCOM configuration, dc voltage control is required to ensure that the rated capacitor
bank voltage is not exceeded nor run down, but maintained at a level which can deliver or absorb
some amount of energy so as to provide a dynamic ride-through capability during transient events
[Rao et al. 2000]. The capacitor also attenuates the harmonic distortion of the dc bus voltage
5.4
CLOSED-LOOP CONTROLLED MODEL CONSTRUCTION
75
from the dc current harmonics as seen in Figs. 5.6 to 5.9. Because of this harmonic distortion,
there must be some provision for filtering the signal when used an an input to the control
system. This distortion may originate from the harmonic currents of the converters switching
action, modulated from the characteristic ac current components (thus giving the characteristic
0 ± 6n components on the dc side, more readily filtered), or a second-harmonic distortion arising
from an imbalance in the fundamental current, which being a low frequency is harder to filter,
being presented a higher impedance by the dc-side capacitor.
Using the technique described above, the real part of the zero frequency component may be
extracted from the dc voltage harmonic vector (the imaginary part will always be zero for
conjugate system inputs and thus real signals), and used as a proxy for the averaged value of
the dc voltage.
vdc (t) = <Vdc,0 (t)
(5.16)
This signal will be free of the harmonic distortions present of the time-domain signal as evaluated
through the harmonic reconstruction of the time series, or as seen in the PSCAD time-domain
simulation.
In this model, a single-pole low-pass filter with a time constant of 0.02 s is interposed between the
voltage measurement and the control block in both the PSCAD and state-space systems. In the
PSCAD model, this filter attenuates the high-frequency content which would be passed through
the proportional elements of the control system, and contribute to jitter of the switching instants
through the PWM process. There is little high-frequency content to filter in the zero-frequency
output from the state-space model, but the same filter element must be present to give the same
transient response to the slowly-varying component.
5.4.1.2
Reactive Power
Most modern VSC control systems use a nested system of feedback controllers, with an internal
fast dq-axis current controller taking its reference input from an external control loop tracking
reactive power and dc voltage setpoints. Measurement of reactive power is complicated by
the contribution of harmonic and unbalanced flows to the instantaneous reactive power, as
supported by the rich literature on power measurement for three-phase systems in the presence
of imbalances and distortions [Depenbrock 1993][Watanabe et al. 1993][Peng and Lai 1996]. The
explicitly balanced HSS formulation permits a powerful simplifying assumption; that the relevant
quantity for support of the grid voltage is the fundamental frequency reactive power flow at the
76
CHAPTER 5
VSC MODEL WITH CONTROL
grid connection point. The complex power, defined in positive sequence fundamental phasors,
yields the following formulation in a stationary dq reference frame.
S = V I∗
(5.17)
= (Vd + jVq )(Id − jIq )
(5.18)
= (Vd Id + Vq Iq ) + j(Vq Id − Vd Iq )
(5.19)
In the special case where the dq reference frame is identical with the grid (and thus V = Vd , Vq =
0), the reactive power component reduces to
=S = −Vd Iq
(5.20)
so that for a constant and normalised Vd , the Iq component functions as a directly proportional
proxy for fundamental frequency reactive power levels, with positive Iq equal to reactive power
delivered under the passive sign convention adopted here. As a small-signal model, linearisation
around a V = Vd , Vq = 0 operating point gives second-order terms for changes in the phase of
the remote source which are ignored.
As the HSS system is defined in a fixed global dq reference frame, the direct and quadrature
components of any fundamental ac signal may be accessed by inspection of the fundamental
frequency phasor, again using the assumption of the quasi-steady state relaxation of the harmonic
state variables. Compared to time-domain analogues of reactive power measurement which use
the full harmonic spectrum, this removes direct dependence on the maximum harmonic order
used in the model, allowing for comparison over varying maximum harmonic orders.
Because the real and imaginary parts of the complex EMP phasors are separate state variables,
this imaginary component operation is implemented in the state-space model simply through
the linear combination of the two conjugate (for real inputs) imaginary variables as given in
equation 5.21. The result is equivalent to the magnitude-preserving form of the dq0 transform
given in 4.3.2, with the steady d and q elements equivalent to peak phase quantities in the ac
system.
iq (t) = =Ia,+1 (t) − =Ia,−1 (t)
(5.21)
5.4
CLOSED-LOOP CONTROLLED MODEL CONSTRUCTION
77
For the time-domain PSCAD model, no such built-in reference frame transformation is available,
and the quadrature current quantity must be calculated by means of a real-time dq transform
block, using the same fixed phase reference from a separated ideal bus as the control system. The
dq transform functions as a linear frequency modulator in a very similar way to the switching
converter itself, down-modulating the balanced fundamental current component to a pair of id
and iq signals. These resultant signals have stationary components from the balanced fundamental component, as well as contributions from down-modulated harmonics and negative-sequence
components.
As for the dc voltage measurement, a single-pole filter with a time constant of 0.02 s is used in
both models to attenuate these distortions in the PSCAD model while giving the same transient
response. As will be demonstrated in Section 5.5.5, if an imbalance were introduced in the HSS
model through the modulation of the grid voltage fundamental phasor at −100 Hz, then the
same 100 Hz modulation would show up in the state-space measurement as that produced from
the down-modulation of the PSCAD dq transform, further validation of the principle of using
dynamic variation of balanced HSS phasors to represent imbalanced signals.
5.4.2 Control System
The intention of this control model is not to emulate a practical industrial control system, but
to serve as a test system to evaluate closed-loop system dynamics using LTI system analysis
techniques. To this end, the control system implemented has a simple feedback structure which
satisfies closed-loop stability and controllability criteria, able to reach a steady-state control
set-point in an acceptable simulation timeframe.
The structure of this control system is shown in Fig. 5.17, with a 2x2 gain matrix translating
control variable errors into control input errors, and then a pair of proportional-integral blocks
to track steady-state reference inputs. The error gain matrix is derived numerically by taking
the inverse of the open-loop sensitivity matrix S described in the simulation procedure of Section
5.4.4, taken once and then fixed to remove dependence on design variables under test such as the
simulation harmonic order. An identical structure is implemented in the PSCAD model through
multiplier and PI blocks.
This control network translates directly to the following state-space form:
78
CHAPTER 5
VSC MODEL WITH CONTROL
Iq−set
Iq−meas
Error Gains
K
Vdc−meas
Kp +
Ki
s
δMd
Kp +
Ki
s
δMq
Vdc−set
Block diagram of feedback network.
Figure 5.17
"
A=
C=
#
(5.22)
0 0
"
B=
0 0
k1,1 k1,2
#
(5.23)
k2,1 k2,2
"
#
Ki 0
0
"
D = Kp
(5.24)
Ki
k1,1 k1,2
k2,1 k2,2
#
(5.25)
5.4.3 Complete Closed-Loop Model
Following the procedure of state-space system construction of Section 4.4.4, the complete closedloop model is constructed from the converter transfer, ac network, dc network, measurement,
filtering and control blocks as illustrated in Fig. 5.18.
The input vector to the closed-loop system consists of the grid voltages plus the control setpoints.
5.4
79
CLOSED-LOOP CONTROLLED MODEL CONSTRUCTION
AC Network
Uncontrolled Transfers
V dc
Va
DC Network
I dc
Ia
Controlled Transfers
∂V a
∂Md
1
1+sTf
∂V a
∂Mq
Iq−set
Error Gains
K
Figure 5.18
∂I dc
∂Md
1
1+sTf
∂I dc
∂Mq
δMq
δMd
Kp +
Ki
s
Kp +
Ki
s
Vdc−set
Block diagram of small-signal closed-loop control model.

..
.






Vgrid,−5 




V

 grid,+1 



Vgrid,+7 






..


 . 


..


. 
U =



V


grid,−7





Vgrid,−1 




V
 grid,+5 



..


.




 Iq−set 


Vdc−set
(5.26)
80
CHAPTER 5
VSC MODEL WITH CONTROL
5.4.4 Simulation Procedure
The procedure for initialising the closed-loop model requires multiple steps. Some of these will
be unique to the particular operating point, and require re-computation for any changes; others,
such as the dynamic inputs, permit repeated simulation with varying values from the state-space
system generated in the prior steps. Aside from the configuration and specification of the passive
network, the closed-loop simulation is defined by a set of fixed parameters and time-series:
1. The maximum harmonic order h, defined at 1 + 6n for ac vectors.
2. The base-case network inputs V grid−base , identified with the first orthogonal mode of
Section 5.2.1, normally a single fundamental frequency energisation.
3. The base-case control setpoint, given as the pair Iq−base and Vdc−base
~ base , chosen to give an uncontrolled steady4. The converter base-case modulation vector M
state response in the vicinity of the base-case control setpoint (it is not necessary to be
exact because the steady-state solution to the closed-loop model will include the adjustment
necessary for convergence to the base-case control setpoint).
5. The dynamic network inputs V grid (t), a time series beginning at the base-case value.
6. The dynamic control setpoint inputs Iq−set (t) and Vdc−set (t), a time series beginning at
the base-case value.
The simulation procedure itself is then as follows:
1. Build the open-loop system (as per Fig. 5.1) with control transfers zeroed.
2. Solve for the steady-state solution to the base-case network inputs by inversion of the
system matrices.
3. Extract the base-case harmonic vectors for the linked converter state variables I a−base and
V dc−base .
4. (Optional) Build the open-loop system with the control transfers incorporating the basecase state variables.
5. (Optional) Solve for the steady-state solution to the base-case inputs plus small perturbations of Md and then Mq to generate the open-loop control sensitivity matrix S as per
(5.27).
6. Build the closed-loop system (as per Fig. 5.18) with the same control transfers.
5.5
81
CLOSED-LOOP SIMULATION RESULTS
7. Solve the closed-loop system for the base-case network and control reference inputs.
8. Simulate the dynamic closed-loop system for the time-varying network and control reference inputs.
"
δIq
δVdc
#
"
=
s1,1 s1,2
#"
#
δMd
s2,1 s2,2
δMq
(5.27)
In the direct solution by matrix inversion of step 7, the closed-loop stability and controllability
of the system is implied by the invertibility of the system A matrix. The system state produced
by this solution has the control block integrators pre-set to the level that gives convergence with
the base-case setpoint, so that the base-case modulation vector does not need to be precisely set
to achieve the base-case control setpoint with no control input.
5.5 CLOSED-LOOP SIMULATION RESULTS
This section presents simulation results obtained using the closed-loop model. Table 5.2 lists
the common system parameters for closed-loop simulation. The switching frequency has been
lowered to mf = 9 so as to make the effect of low-order harmonic interaction more apparent.
The base control gains were selected to give a stable but underdamped response.
The source of the measured Iq plotted in both the HSS and EMTDC data is the output of
the single-pole filter, which forms the input to the control system. This allows the transient
response to be visually matched between the simulations, although the harmonic content of the
time-domain Iq measurement has not been completely attenuated, so the comparison must be
performed with reference to the envelope of the signal.
5.5.1 Controlled Model Setpoint Tracking
In the first set of simulations, a step change of the Iq control setpoint from 215 A to 220 A was
ordered at t = 100 ms. Figs. 5.19 - 5.21 demonstrate the close time-domain fit for the small
change in the quadrature current setpoint, for maximum harmonic order of the HSS simulation
h = 49.
With a larger deviation of the setpoint from the base case at Iq = 250 A, the limits of the linearised model are visibly encountered. Figs. 5.22 and 5.23 show the divergence of the dominant
closed-loop mode from that of the time-domain simulation, and Fig. 5.24 shows the time series
of the δMd and δMq control inputs to the linearised model.
82
CHAPTER 5
VSC MODEL WITH CONTROL
Quadrature Current Following Iq Setpoint Change to 0.220 kA
0.223
0.222
0.221
Current (kA)
0.22
0.219
0.218
0.217
0.216
0.215
HSS
EMTDC
0.214
0
100
Figure 5.19
200
300
400
500
Time (ms)
600
700
800
900
1000
Quadrature converter current following a change in the Iq setpoint.
DC Voltage Following Iq Setpoint Change to 0.220 kA
30.15
Voltage (kV)
30.1
30.05
30
29.95
HSS
EMTDC
29.9
0
100
200
Figure 5.20
300
400
500
Time (ms)
600
700
800
DC bus voltage following a change in the Iq setpoint.
900
1000
83
CLOSED-LOOP SIMULATION RESULTS
DC Voltage Following Iq Setpoint Change to 0.220 kA
30
Voltage (kV)
29.98
29.96
29.94
29.92
HSS
EMTDC
29.9
100
Figure 5.21
105
110
115
120
125
Time (ms)
130
135
140
145
150
DC bus voltage following a change in the Iq setpoint, detailing initial transient response.
0.26
0.255
0.25
0.245
0.24
Current (kA)
5.5
0.235
0.23
0.225
0.22
0.215
HSS
EMTDC
0.21
0
100
Figure 5.22
200
300
400
500
Time (ms)
600
700
800
900
1000
Quadrature converter current following a larger change in the Iq setpoint.
84
CHAPTER 5
VSC MODEL WITH CONTROL
DC Voltage Following Iq Setpoint Change to 0.250 kA
30.3
30.25
30.2
30.15
Voltage (kV)
30.1
30.05
30
29.95
29.9
29.85
HSS
EMTDC
29.8
0
100
200
Figure 5.23
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following a larger change in the Iq setpoint.
Applied Control Inputs Following Iq Setpoint Change to 0.250 kA
0.01
0.005
Control Input (p.u.)
0
−0.005
−0.01
−0.015
−0.02
−0.025
Md
Mq
−0.03
0
100
Figure 5.24
200
300
400
500
Time (ms)
600
700
800
900
Applied control inputs following a larger change in the Iq setpoint.
1000
5.5
85
CLOSED-LOOP SIMULATION RESULTS
Table 5.2
Closed-loop test system parameters.
Parameter
Value
Base voltage
Base MVA
Fundamental frequency
Switching frequency
Base modulation vector
Source resistance
Source inductance
Converter resistance
Converter inductance
Filter resistance
Filter capacitance
DC-side capacitance
DC-side resistance
Converter LC corner frequency
Quadrature current base setpoint
DC voltage base setpoint
Proportional gain
Integral gain
Vbase = 10 kV
Sbase = 2 MVA
f0 = 50 Hz
fsw = 450 Hz
~ base = 0.8 − j0.03
M
Rs = 0.25 Ω
Ls = 0.01 H
Rc = 1 Ω
Lc = 0.04 H
Rf = 0.1 Ω
Cf = 5 µF
Cdc = 300 µF
Rdc = 10 kΩ
fco = 355.881 Hz
Iq−base = 215 A (peak)
Vdc−base = 30 kV
Kp = 0.5
Ki =20
−0.2 −10
S=
−50 −300
0.6818 −0.0227
−1
K=S =
−0.1136 0.0005
Perturbation matrix (rounded)
Control gain matrix
Per-unit basis
9 p.u.
0.005 p.u.
0.0628 p.u.
0.02 p.u.
0.251 p.u.
0.002 p.u.
12.7 p.u.
0.212 p.u.
2000 p.u.
7.12 p.u.
1.31 p.u.
5.5.2 Response to Grid Perturbations
To further test the limits of the linearised model, the response to a drop in grid voltage is simulated, an important response for an actual STATCOM. The control parameters were specified to
be more oscillatory, at Kp = 0.8, Ki = 50. As evidenced in Figs. 5.25-5.28, the 1% grid voltage
step kept the model within the same closely linear envelope as the small Iq setpoint changes,
but the 5% step showed up a divergence in the dominant oscillatory mode.
86
CHAPTER 5
VSC MODEL WITH CONTROL
Quadrature Current Following 1% Grid Voltage Drop
0.22
0.219
0.218
Current (kA)
0.217
0.216
0.215
0.214
0.213
HSS
EMTDC
0.212
0
100
Figure 5.25
200
300
400
500
Time (ms)
600
700
800
900
1000
Quadrature converter current following a 1% step reduction in the grid voltage.
DC Voltage Following 1% Grid Voltage Drop
30.25
30.2
30.15
Voltage (kV)
30.1
30.05
30
29.95
29.9
HSS
EMTDC
29.85
0
100
Figure 5.26
200
300
400
500
Time (ms)
600
700
800
900
DC bus voltage following a 1% step reduction in the grid voltage.
1000
87
CLOSED-LOOP SIMULATION RESULTS
Quadrature Current Following 5% Grid Voltage Drop
0.23
0.225
Current (kA)
0.22
0.215
0.21
0.205
HSS
EMTDC
0.2
0
100
Figure 5.27
200
300
400
500
Time (ms)
600
700
800
900
1000
Quadrature converter current following a 5% step reduction in the grid voltage.
DC Voltage Following 5% Grid Voltage Drop
30.4
30.3
30.2
30.1
Voltage (kV)
5.5
30
29.9
29.8
29.7
29.6
HSS
EMTDC
29.5
0
100
Figure 5.28
200
300
400
500
Time (ms)
600
700
800
900
DC bus voltage following a 5% step reduction in the grid voltage.
1000
88
CHAPTER 5
VSC MODEL WITH CONTROL
5.5.3 Model Order and Harmonic Interaction
The design of the HSS model allows for the independent variation of the highest modelled
harmonic order h in the simulation, with all harmonic state variables and associated coupling
terms above this harmonic frequency being neglected. This permits a unique form of experiment,
in which the same simulation is evaluated for different harmonic orders to test for the effects of
coupling at harmonic frequencies.
The simulation of an Iq setpoint change in Section 5.5.1 was evaluated for h = 1, 7, 13, 19, 49,
with the limiting case of h = 1 being the fundamental frequency only model. The results
clustered very tightly into two groups at h = 1, 7, 13 and h = 19, 49. Figs. 5.29 and 5.30 show,
using the state-space outputs of the models, that the dominant mode of the closed-loop response
is dependent on harmonic coupling effects at h = 19 and below. Fig. 5.31 demonstrates that no
further significant change in the response is caused by an increase in the harmonic order from
19 to 49.
Given that the higher-order models are able to accurately match the output of the time-domain
simulation in PSCAD as evidenced in Section 5.5.1, it can be concluded that the fundamental
frequency model lacks the ability to fully describe the closed-loop dynamics of the converter
system with harmonic coupling, although it will be sufficiently close for basic control analysis
in isolation. This manner of dependence on harmonic order is expected to be important when
evaluating models incorporating multiple converters or other harmonic sources.
5.5
89
CLOSED-LOOP SIMULATION RESULTS
Quadrature Current Following Iq Setpoint Change to 0.220 kA
0.222
0.221
0.22
Current (kA)
0.219
0.218
0.217
0.216
0.215
HSS (h=19)
HSS (h=13)
0.214
0
100
200
300
400
500
Time (ms)
600
700
800
900
1000
Figure 5.29 Quadrature converter current following a change in the Iq setpoint, showing divergent response
for varied maximum harmonic order.
DC Voltage Following Iq Setpoint Change to 0.220 kA
30.02
30.015
30.01
Voltage (kV)
30.005
30
29.995
29.99
29.985
HSS (h=19)
HSS (h=13)
29.98
0
100
200
300
400
500
Time (ms)
600
700
800
900
1000
Figure 5.30 DC bus voltage following a change in the Iq setpoint, showing divergent response for varied
maximum harmonic order.
90
CHAPTER 5
VSC MODEL WITH CONTROL
DC Voltage Following Iq Setpoint Change to 0.220 kA
30.015
30.01
Voltage (kV)
30.005
30
29.995
29.99
29.985
HSS (h=19)
HSS (h=49)
29.98
100
120
140
160
180
200
Time (ms)
220
240
260
280
300
Figure 5.31 DC bus voltage following a change in the Iq setpoint, showing close agreement for further increases
in harmonic order beyond h=19.
91
CLOSED-LOOP SIMULATION RESULTS
Pole−Zero Map
2000
1500
1000
Imaginary Axis (seconds−1)
5.5
500
0
−500
−1000
−1500
−2000
−20
−15
−10
−5
−1
Real Axis (seconds )
Figure 5.32
Pole-zero map of closed-loop system for h=1 (red), h=19 (blue).
92
CHAPTER 5
VSC MODEL WITH CONTROL
5.5.4 Control Parameter Root Loci
With the basic validity of the closed-loop model established, and the demonstrated link between
the identified dominant system poles and the system response, the natural application is the
analysis of the control system through the generation of root-loci for variation in the control
parameters. These root-loci are unique to the operating point that the closed-loop system was
linearised around, and must be approximated by a discrete series of pole-zero plots with varied
parameters.
The control system as implemented has two free parameters, the proportional and integral gain
constants Kp and Ki , applied to the output of the fixed gain matrix. Figs. 5.36 and 5.37 show
the locus for Kp , with the movement of the dominant pole-pair to the left implying a faster
settling time as the gain increases, and the other significant group of poles moving right. Figs.
5.38 shows a similar locus for the integral gain, but with the dominant poles becoming both
slower and more oscillatory.
The time-domain responses for these parameters are shown in Figs. 5.33 and 5.34, confirming
the interpretation of the dominant poles. Fig. 5.35 gives the response of the controller tuned
with these root-loci to give a quick settling time to the new setpoint, particularly the current
demand, using Kp = 1.0, Ki = 10, and Fig. 5.39 gives its pole-zero map.
DC Voltage Following Iq Setpoint Change to 0.220 kA for Varying Kp
30.025
30.02
30.015
30.01
Voltage (kV)
30.005
30
29.995
29.99
29.985
Kp=0.5
Kp=0.6
Kp=0.7
29.98
29.975
0
Figure 5.33
100
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following a change in the Iq setpoint, plotted for different values of Kp .
5.5
93
CLOSED-LOOP SIMULATION RESULTS
DC Voltage Following Iq Setpoint Change to 0.220 kA for Varying Ki
30.03
30.02
Voltage (kV)
30.01
30
29.99
29.98
Ki=20
Ki=30
Ki=40
29.97
0
Figure 5.34
100
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following a change in the Iq setpoint, plotted for different values of Ki .
30.02
0.22
30.01
0.218
30
0.216
29.99
0.214
0
200
400
600
800
1000
Time (ms)
1200
1400
1600
1800
DC Voltage (kV)
Quadrature−Axis Current (kA)
Filtered Control Variables Following Iq Setpoint Change to 0.220 kA with Tuned Control Parameters
0.222
29.98
2000
Figure 5.35 DC bus voltage and quadrature current following a change in the Iq setpoint, demonstrating fast
settling with a properly tuned controlled.
94
CHAPTER 5
VSC MODEL WITH CONTROL
Pole−Zero Map
4
1.5
x 10
1
Imaginary Axis (seconds−1)
0.5
0
−0.5
−1
−1.5
−70
−60
−50
−40
−30
−20
−10
0
Real Axis (seconds−1)
Figure 5.36
Pole-zero map of closed-loop system for Kp = 0.5 (blue), Kp = 0.6 (green), Kp = 0.7 (red).
5.5
95
CLOSED-LOOP SIMULATION RESULTS
Pole−Zero Map
600
400
Imaginary Axis (seconds−1)
200
0
−200
−400
−600
−16
−14
−12
−10
−8
−6
−4
−2
0
Real Axis (seconds−1)
Figure 5.37 Pole-zero map of closed-loop system for Kp = 0.5 (blue), Kp = 0.6 (green), Kp = 0.7 (red),
detailing locus of dominant poles.
96
CHAPTER 5
VSC MODEL WITH CONTROL
Pole−Zero Map
600
400
−1
Imaginary Axis (seconds )
200
0
−200
−400
−600
−16
−14
−12
−10
−8
−6
−4
−2
Real Axis (seconds−1)
Figure 5.38 Pole-zero map of closed-loop system for Ki = 20 (blue), Ki = 30 (green), Ki = 40 (red), detailing
locus of dominant poles.
97
CLOSED-LOOP SIMULATION RESULTS
Pole−Zero Map
600
400
200
Imaginary Axis (seconds−1)
5.5
0
−200
−400
−600
−18
−16
−14
−12
−10
−8
−6
−4
−2
Real Axis (seconds−1)
Figure 5.39
Pole-zero map of tuned closed-loop system, detailing dominant poles.
98
CHAPTER 5
VSC MODEL WITH CONTROL
5.5.5 Closed-Loop Response to Grid Imbalance
The closed-loop response to a grid imbalance, in the form of an introduced negative sequence
fundamental component, is a fundamental test of the entire modelling framework introduced in
this thesis. Using the modulated EMP phasor technique described earlier, a 5% imbalance was
introduced to the grid voltage at t = 100 ms to the STATCOM system with constant control
setpoints and using the tuned control parameters of the previous section. As seen in Figs. 5.405.44, the model captures both the envelope of the closed-loop response, and at h = 49, the fine
time-domain features. In Fig. 5.40, the current measurement can be eventually seen to relax to
the pure 100 Hz excitation.
As discussed in Section 5.4.1.2, using the modulation of a balanced phasor to achieve representation of non-characteristic signals necessarily results in the same modulation being transferred
through to the control system in the same manner as the linear modulation characteristic of the
time-domain dq transform. Using a single harmonic component, in this case the fundamental
component of the current harmonic vector, breaks the equivalence of all dynamic EMP signal
representations. The result is that to be properly reflected in the control measurements, all such
non-characteristic system disturbances must be achieved through modulation of the fundamental frequency component only, in contrast with the uncontrolled model. This is no matter in
practise, as any number of non-characteristic signals can be modulated in superposition, this
also ensuring an absence of direct dependence on the system harmonic order.
Quadrature Current Following 5% Grid Imbalance
0.222
0.22
Current (kA)
0.218
0.216
0.214
0.212
0.21
HSS
EMTDC
0.208
0
100
Figure 5.40
200
300
400
500
Time (ms)
600
700
800
900
1000
Quadrature converter current following the introduction of a 5% imbalance.
5.5
99
CLOSED-LOOP SIMULATION RESULTS
Quadrature Current Following 5% Grid Imbalance
0.222
0.22
Current (kA)
0.218
0.216
0.214
0.212
0.21
HSS
EMTDC
0.208
100
120
140
160
Time (ms)
180
200
220
240
Figure 5.41 Quadrature converter current following following the introduction of a 5% imbalance, detailing
initial transient response.
Applied Control Inputs Following 5% Grid Imbalance
−0.024
D−Axis Control Input (p.u.)
−0.026
−0.028
−0.03
−0.032
−0.034
Md (HSS)
Md (EMTDC)
−0.036
Figure 5.42
response.
100
120
140
160
Time (ms)
180
200
220
240
D-axis control signal following the introduction of a 5% imbalance, detailing initial transient
100
CHAPTER 5
VSC MODEL WITH CONTROL
DC Voltage Following 5% Grid Imbalance
30.3
30.2
Voltage (kV)
30.1
30
29.9
29.8
29.7
HSS
EMTDC
29.6
0
100
Figure 5.43
200
300
400
500
Time (ms)
600
700
800
900
1000
DC bus voltage following the introduction of a 5% imbalance.
DC Voltage Following 5% Grid Imbalance
30.3
30.2
Voltage (kV)
30.1
30
29.9
29.8
29.7
HSS
EMTDC
29.6
Figure 5.44
100
120
140
160
Time (ms)
180
200
220
240
DC bus voltage following the introduction of a 5% imbalance, detailing initial transient response.
5.6
CONCLUSIONS
101
5.6 CONCLUSIONS
This chapter presented the extension of the fixed-switching HSS VSC model to a closed-loop
controlled model using a dual linearisation of the numerical switching function and converter
state variables. Simulation results evaluated against PSCAD/EMTDC modelling demonstrate
the validity of the linearised model, judged by the ability to accurately capture closed-loop
dynamics within a small-signal region of interest that was empirically probed. As demonstrated
by the use of generated root-loci to tune the control system, LTI system analysis tools can be
meaningfully applied to the linearised HSS model.
One interesting result was the relationship between the modelled harmonic order and the closedloop response, which revealed a sharp discontinuity at h = 19. This demonstrates the flexibility
of the fully general HSS modelling approach, which does not require a-priori knowledge of which
terms will be significant. The other major result is the validation of the reduced-order modelling
framework, proving that unbalanced three-phase closed-loop system dynamics can be effectively
captured while preserving major computational efficiency gains.
Chapter 6
CONCLUSIONS AND FUTURE WORK
This final chapter summarises the conclusions drawn from the results obtained, and those formed
in the process of developing the model. Directions for further research leading from this work
are then discussed.
6.1 CONCLUSIONS
Model Applicability
The ideal application for the developed model is the characterisation of low-frequency harmonic
interactions in controlled converters. This is supported by several properties of the model:
• Harmonic flows and couplings can be identified directly in the model output.
• The maximum harmonic order can be easily selected, with no a-priori knowledge of the
possible frequency range of interest required.
• The response to closed-loop control can be accurately modelled in a small-signal region of
interest around a specified operating point.
• The modular approach to system construction allows for many permutations of VSC system
to be easily constructed.
• Fast simulation times permit rapid iteration of design variables and operating points.
The simulations in this thesis use switching frequencies of 450 Hz to 750 Hz (mf = 9 to mf = 15)
so as to maximise observed harmonic interaction in a single-converter system. This corresponds
to the frequency ranges used in [Möllerstedt and Bernhardsson 2000] (where f0 =
50
3 Hz, fsw
=
250 Hz, mf = 15), albeit for a single-phase traction system. This method therefore lends itself to
the analysis of VSC systems using GTO/IGCT devices, which operate generally in the sub-1 kHz
range.
104
CHAPTER 6
CONCLUSIONS AND FUTURE WORK
Modelling Approach
The construction of the converter and associated control system in a global synchronous reference
frame provided an ideal basis for the extension of the converter model to include the effects of
control variation. The numerical switching function model in this reference frame was ideal for
linearisation due to the maximal decoupling between the quadrature control inputs.
The modular system construction method links the electrical and control systems through the
identity created by the balanced fundamental frequency reference frame. By breaking the symmetry of the HSS, this fulfils the goal of unifying the electrical and control analyses, allowing
direct solution for balanced steady-state inputs including control action, and dynamic simulation
for arbitrary inputs including non-characteristic harmonics and other unbalanced components.
State-Space Analysis
State-space analysis methods were found to generalise well to the transformed LTP model of the
VSC. Discrete pole-zero root-locus techniques can provide qualitative and quantitative insight
into the closed-loop system performance.
Model Limitations
One significant limitation is that the control system must be linear. This requires either the
modelled control system to be linear, or a further linearisation to be carried out. Some forms
of non-linearities in control systems such as saturation and anti-windup mechanisms are not
usually relevant in small-signal studies, some such as multiplicative terms are readily linearised,
and others such as variable-structure techniques are not linearisable in this way.
Due to the reduction of the harmonic components of the system to the balanced characteristic
harmonic set, the steady-state solution must be balanced; responses to unbalanced inputs may
not be solved directly, and unbalanced components may not form the base-case for control
linearisation. This is considered to be a feature of the model, as it makes clear the distinction
between the balanced steady state and the arbitrary dynamic variation potentially including
non-characteristic harmonics or even inter-harmonics. The ac network is restricted to balanced
three-phase impedances.
6.2 FUTURE WORK
Paralleled Converter Modelling
With no modification to the developed model, the modular nature of the system construction
should allow paralleled converter systems to be studied. This is a rich domain of harmonic
interactions which are only obliquely analysable using time-domain models.
6.2
FUTURE WORK
105
Extension to Other Topologies
As discussed, the frequency range of significant harmonic coupling effects lends itself to analysis of high-power converters using IGCT devices. To increase the relevance to practical systems using these devices, the next logical step would be to extend the model to three-level
Neutral Point Clamped converters; a valuable topology for IGCT devices [Bhattacharya and
Babaei 2011]. This could be readily accomplished using the numerical switching model developed in this thesis, assuming that the neutral-point balancing could be abstracted away as the
dead-time compensation is here.
VSC-HVDC Modelling
In combination with the extension to multi-level converters, the HSS converter model could be
used to model VSC-HVDC systems, including harmonic interaction on the dc link. Madrigal
and Acha used an uncontrolled harmonic domain PWM model to investigate dc-side harmonic
resonance phenomena [Madrigal and Acha 2001], which could be expanded upon using a controlled dynamic model. There is also scope for the extension of the HSS model to include
inter-harmonic components as in [Ramirez 2011] for the modelling of HVDC systems linking ac
networks of varying frequency.
Integration with Existing HSS Models
HSS models for other FACTS devices and power system components exist, such as the TCR
model developed in [Orillaza 2011] and the HVDC model in [Hwang and Wood 2012]. Future
work could be oriented towards consolidating these models in a unified framework, as these
models differ slightly in their approach, particularly when conceptualising the modelling of threephase systems in the HSS. The challenge will be to consolidate line-commutated switching models
in a modular reduced-order framework, which should be assisted by the balanced harmonic signal
and network representations described in this thesis.
High-Level System Description and Simulation Environment
If the goal of integration and harmonisation of HSS models described above could be achieved,
an ideal next step would be the creation of a generalised simulation framework. The potentially
laborious generation of models could be automated, by means of a high-level schema describing
the topology and parameters of the system as interconnected blocks, and an associated interpreter to build the state-space system and collate its output. This would be an ambitious step
towards a general-purpose HSS simulation environment.
REFERENCES
Alvarez Martin, J.A., Melgoza, J.R. and Rincon Pasaye, J.J. (2011), ‘Exact steady
state analysis in power converters using Floquet decomposition’, In North American Power
Symposium (NAPS), 2011, IEEE, August, pp. 1–7.
Arrillaga, J. and Watson, N.R. (2008), ‘The Harmonic Domain revisited’, In Harmonics
and Quality of Power, 2008. ICHQP 2008. 13th International Conference on, IEEE, pp. 1–
9.
Arrillaga, J., Medina, A., Lisboa, Cavia, M.A. and Sanchez, P. (1995), ‘The Harmonic
Domain. A frame of reference for power system harmonic analysis’, Power Systems, IEEE
Transactions on, Vol. 10, No. 1, February, pp. 433–440.
Bhattacharya, S. and Babaei, S. (2011), ‘Series connected IGCT based three-level Neutral Point Clamped voltage source inverter pole for high power converters’, In Energy
Conversion Congress and Exposition (ECCE), 2011 IEEE, IEEE, pp. 4248–4255.
Bracewell, R. (2000), The Fourier transform and its applications, McGraw-Hill series in
electrical and computer engineering, McGraw Hill.
Chavez, J.D. (2010), ‘A modified dynamic harmonic domain distribution line model’, In Power
and Energy Society General Meeting, 2010 IEEE, IEEE, July, pp. 1–7.
Collins, C.D. (2006), FACTS device modelling in the harmonic domain, PhD thesis, University
of Canterbury, Christchurch, New Zealand.
Deore, S.R., Darji, P.B. and Kulkarni, A.M. (2012), ‘Dynamic phasor modeling of Modular Multi-level Converters’, In Industrial and Information Systems (ICIIS), 2012 7th IEEE
International Conference on, IEEE, August, pp. 1–6.
Depenbrock, U. (1993), ‘The FBD-method, a generally applicable tool for analyzing power
relations’, Power Systems, IEEE Transactions on, Vol. 8, No. 2, May, pp. 381–387.
Dommel, H.W. (1969), ‘Digital Computer Solution of Electromagnetic Transients in Single-and
Multiphase Networks’, Power Apparatus and Systems, IEEE Transactions on, Vol. PAS-88,
No. 4, April, pp. 388–399.
108
REFERENCES
Etxeberria-Otadui, I., Viscarret, U., Caballero, M., Rufer, A.C. and Bacha, S.
(2007), ‘New Optimized PWM VSC Control Structures and Strategies Under Unbalanced
Voltage Transients’, Industrial Electronics, IEEE Transactions on, Vol. 54, No. 5, October,
pp. 2902–2914.
Fang, C.C.C. and Abed, E.H. (1999), ‘Sampled-data modeling and analysis of closed-loop
PWM DC-DC converters’, In Circuits and Systems, 1999. ISCAS &#039;99. Proceedings
of the 1999 IEEE International Symposium on, IEEE, pp. 110–115 vol.5.
Floquet, G. (1883), ‘Sur les équations différentielles linéairesa coefficients périodiques’, Ann.
Sci. École Norm. Sup, Vol. 12, pp. 47–88.
Fortescue, C.L. (1918), ‘Method of Symmetrical Co-Ordinates Applied to the Solution of
Polyphase Networks’, American Institute of Electrical Engineers, Transactions of the,
Vol. XXXVII, No. 2, July, pp. 1027–1140.
Hill, G.W. (1886), ‘On the part of the motion of the lunar perigee which is a function of the
mean motions of the sun and moon’, Vol. 8, No. 1, pp. 1–36.
Holmes, D.G. (1998), ‘A general analytical method for determining the theoretical harmonic
components of carrier based PWM strategies’, In Industry Applications Conference, 1998.
Thirty-Third IAS Annual Meeting. The 1998 IEEE, IEEE, October, pp. 1207–1214 vol.2.
Holmes, D. and Lipo, T. (2003), Pulse width modulation for power converters: principles and
practice, IEEE Press series on power engineering, IEEE Press.
Hume, D.J., Wood, A.R. and Osauskas, C.M. (2003), ‘Frequency-domain modelling
of interharmonics in HVDC systems’, Generation, Transmission and Distribution, IEE
Proceedings-, Vol. 150, No. 1, January, pp. 41–48.
Hwang, M.S. and Wood, A.R. (2012), ‘Harmonic State-Space modelling of an HVdc converter’, In Harmonics and Quality of Power (ICHQP), 2012 IEEE 15th International
Conference on, IEEE, June, pp. 573–580.
Kolar, J.W., Ertl, H. and Zach, F.C. (1991), ‘Influence of the modulation method on the
conduction and switching losses of a PWM converter system’, Industry Applications, IEEE
Transactions on, Vol. 27, No. 6, November, pp. 1063–1075.
Larsen, E.V., Baker, D.H. and McIver, J.C. (1989), ‘Low-order harmonic interactions on
AC/DC systems’, Power Delivery, IEEE Transactions on, Vol. 4, No. 1, January, pp. 493–
501.
Love, G.N. (2007), Small Signal Modelling of Power Electronic Converters for the Study
of Time-domain Waveforms, PhD thesis, University of Canterbury, Christchurch, New
Zealand.
109
Madrigal, M. and Acha, E. (2001), ‘Harmonic modelling of voltage source converters for
HVDC stations’, In AC-DC Power Transmission, 2001. Seventh International Conference
on (Conf. Publ. No. 485), IET, November, pp. 125–131.
Madrigal, M. and Rico, J.J. (2004), ‘Operational Matrices for the Analysis of Periodic Dynamic Systems’, IEEE Transactions on Power Systems, Vol. 19, No. 3, August, pp. 1693–
1695.
Madrigal, M., Acha, E., Mayordomo, J.G., Asensi, R. and Hernandez, A. (2000),
‘Single-phase PWM converters array for three-phase reactive power compensation. II. Frequency domain studies’, In Harmonics and Quality of Power, 2000. Proceedings. Ninth
International Conference on, IEEE, pp. 645–651 vol.2.
Mathur, R. and Varma, R. (2002), Thyristor-Based FACTS Controllers for Electrical Transmission Systems, IEEE Press series on microelectronic systems, Wiley.
Mohan, N. and Undeland, T. (2007), Power electronics: converters, applications, and design,
Wiley India.
Möllerstedt, E. and Bernhardsson, B.M. (2000), ‘Out of control because of harmonicsan analysis of the harmonic response of an inverter locomotive’, Control Systems, IEEE,
Vol. 20, No. 4, August, pp. 70–81.
Orillaza, J.R.C. (2011), Harmonic State Space Model of Three Phase Thyristor Controlled
Reactor, PhD thesis, University of Canterbury, Christchurch, New Zealand.
Orillaza, J.R., Hwang, M.S. and Wood, A.R. (2010), ‘Switching Instant Variation in
Harmonic State-Space modelling of power electronic devices’, In Universities Power Engineering Conference (AUPEC), 2010 20th Australasian, IEEE, December, pp. 1–5.
Peng, F.Z. and Lai, J.S. (1996), ‘Generalized instantaneous reactive power theory for threephase power systems’, Instrumentation and Measurement, IEEE Transactions on, Vol. 45,
No. 1, February, pp. 293–297.
Pöllänen, R. and Others (2003), ‘Converter-flux-based current control of voltage source
PWM rectifiers-analysis and implementation’, Acta Universitatis Lappeenrantaensis ISSN
1456-4491.
Ramirez, A. (2011), ‘The Modified Harmonic Domain: Interharmonics’, Power Delivery, IEEE
Transactions on, Vol. 26, No. 1, January, pp. 235–241.
Rao, P., Crow, M.L. and Yang, Z. (2000), ‘STATCOM control for power system voltage control applications’, Power Delivery, IEEE Transactions on, Vol. 15, No. 4, October,
pp. 1311–1317.
110
REFERENCES
Rico, J.J., Madrigal, M. and Acha, E. (2003), ‘Dynamic harmonic evolution using the
extended harmonic domain’, Power Delivery, IEEE Transactions on, Vol. 18, No. 2, April,
pp. 587–594.
Sandberg, H., Möllerstedt, E. and Bernhardsson (2005), ‘Frequency-domain analysis of
linear time-periodic systems’, Automatic Control, IEEE Transactions on, Vol. 50, No. 12,
December, pp. 1971–1983.
Sao, C.K., Lehn, P.W., Iravani, M.R. and Martı́nez, J.A. (2002), ‘A benchmark system for digital time-domain simulation of a pulse-width-modulated D-STATCOM’, Power
Delivery, IEEE Transactions on, Vol. 17, No. 4, October, pp. 1113–1120.
Stanković, A.M., Mattavelli, P., Caliskan, V.A. and Verghese, G.C. (2000), ‘Modeling and analysis of FACTS devices with dynamic phasors’, In Power Engineering Society
Winter Meeting, 2000. IEEE, IEEE, pp. 1440–1446 vol.2.
Vajta, M. (2000), ‘Some remarks on padé-approximations’, In Proc. 3rd TEMPUSINTCOM
Symp. Intelligent Systems in Control and Measurement, pp. 53–58.
Watanabe, E.H., Stephan, R.M. and Aredes, M. (1993), ‘New concepts of instantaneous
active and reactive powers in electrical systems with generic loads’, Power Delivery, IEEE
Transactions on, Vol. 8, No. 2, April, pp. 697–703.
Wereley, N.M. (1991), Analysis and Control of Linear Periodically Time Varying Systems,
PhD thesis, M.I.T.
Wood, A.R. and Osauskas, C.M. (2004), ‘A linear frequency-domain model of a STATCOM’,
Power Delivery, IEEE Transactions on, Vol. 19, No. 3, July, pp. 1410–1418.
Zhou, K. and Wang, D.W. (2002), ‘Relationship between space-vector modulation and threephase carrier-based PWM: a comprehensive analysis’, Industrial Electronics, IEEE Transactions on, Vol. 49, No. 1, February, pp. 186–196.